system analysis via integral quadratic constraints this paper introduces a unified approach to robustness analysis with respect to nonlinearities time variations and uncertain parameters from an original idea by yakubovich 1967 the approach has been developed under a combination of influences from the western and russian traditions of control theory it is shown how a complex system can be described using integral quadratic constraints iqc for its elementary components a stability theorem for systems described by iqcs is presented that covers classical passivity dissipativity arguments but simplifies the use of multipliers and the treatment of causality a systematic computational approach is described and relations to other methods of stability analysis are discussed last but not least the paper contains a summarizing list of iqcs for important types of system components

simulation of confined magnetohydrodynamic flows with dirichlet boundary conditions using a pseudo spectral method with volume penalization a volume penalization approach to simulate magnetohydrodynamic mhd flows in confined domains is presented here the incompressible visco resistive mhd equations are solved using parallel pseudo spectral solvers in cartesian geometries the volume penalization technique is an immersed boundary method which is characterized by a high flexibility for the geometry of the considered flow in the present case it allows to use other than periodic boundary conditions in a fourier pseudo spectral approach the numerical method is validated and its convergence is assessed for two and three dimensional hydrodynamic hd and mhd flows by comparing the numerical results with results from literature and analytical solutions the test cases considered are two dimensional taylor couette flow the z pinch configuration three dimensional orszag tang flow ohmic decay in a periodic cylinder three dimensional taylor couette flow with and without axial magnetic field and three dimensional hartmann instabilities in a cylinder with an imposed helical magnetic field finally we present a magnetohydrodynamic flow simulation in toroidal geometry with non symmetric cross section and imposing a helical magnetic field to illustrate the potential of the method

an immersed boundary method with direct forcing for the simulation of particulate flows we present an improved method for computing incompressible viscous flow around suspended rigid particles using a fixed and uniform computational grid the main idea is to incorporate peskin s regularized delta function approach acta numerica 11 2002 1 into a direct formulation of the fluid solid interaction force in order to allow for a smooth transfer between eulerian and lagrangian representations while at the same time avoiding strong restrictions of the time step this technique was implemented in a finite difference and fractional step context a variety of two and three dimensional simulations are presented ranging from the flow around a single cylinder to the sedimentation of 1000 spherical particles the accuracy and efficiency of the current method are clearly demonstrated

j spectral factorization and equalizing vectors abstract this note presents some basic results on j spectral factorization and its connection with state space realizations the assumptions needed are mild and the proofs are given in detail

brief paper uniqueness conditions for the affine open loop linear quadratic differential game in this note we consider the open loop nash linear quadratic differential game with an infinite planning horizon the performance function is assumed to be indefinite and the underlying system affine we derive both necessary and sufficient conditions under which this game has a unique nash equilibrium

krylov methods for the incompressible navier stokes equations methods are presented for time evolution steady state solving and linear stability analysis for the incompressible navier stokes equations at low to moderate reynolds numbers the methods use krylov subspaces constructed by the arnoldi process from actions of the explicit navier stokes right hand side and of its jacobian without inversion of the viscous operator time evolution is performed by a nonlinear extension of the method of exponential propagation steady states are calculated by inexact krylov newton iteration using orthores and gmres linear stability analysis is carried out using an implicitly restarted arnoldi process with implicit polynomial filters a detailed implementation is described for a pseudospectral calculation of the stability of taylor vortices with respect to wavy vortices in the couette taylor problem 9 1994 academic press inc

a general fictitious domain method with immersed jumps and multilevel nested structured meshes this study addresses a new fictitious domain method for elliptic problems in order to handle general and possibly mixed embedded boundary conditions e b c robin neumann and dirichlet conditions on an immersed interface the main interest of this fictitious domain method is to use simple structured meshes possibly uniform cartesian nested grids which do not generally fit the interface but define an approximate one a cell centered finite volume scheme with a non conforming structured mesh is derived to solve the set of equations with additional algebraic transmission conditions linking both flux and solution jumps through the immersed approximate interface hence a local correction is devised to take account of the relative surface ratios in each control volume for the robin or neumann boundary condition then the numerical scheme conserves the first order accuracy with respect to the mesh step this opens the way to combine the e b c method with a multilevel mesh refinement solver to increase the precision in the vicinity of the interface such a fictitious domain method is very efficient the l 2 and l norm errors vary like o h l where h l is the grid step of the finest refinement level around the interface until the residual first order discretization error of the non refined zone is reached the numerical results reported here for convection diffusion problems with dirichlet robin and mixed dirichlet and robin boundary conditions confirm the expected accuracy as well as the performances of the present method

numerical simulation of fluid structure interaction with the volume penalization method we present a novel scheme for the numerical simulation of fluid structure interaction problems it extends the volume penalization method a member of the family of immersed boundary methods to take into account flexible obstacles we show how the introduction of a smoothing layer physically interpreted as surface roughness allows for arbitrary motion of the deformable obstacle the approach is carefully validated and good agreement with various results in the literature is found a simple one dimensional solid model is derived capable of modeling arbitrarily large deformations and imposed motion at the leading edge as it is required for the simulation of simplified models for insect flight the model error is shown to be small while the one dimensional character of the model features a reasonably easy implementation the coupled fluid solid interaction solver is shown not to introduce artificial energy in the numerical coupling and validated using a widely used benchmark we conclude with the application of our method to models for insect flight and study the propulsive efficiency of one and two wing sections

convolution based interpolation for fast high quality rotation of images this paper focuses on the design of fast algorithms for rotating images and preserving high quality the basis for the approach is a decomposition of a rotation into a sequence of one dimensional translations as the accuracy of these operations is critical we introduce a general theoretical framework that addresses their design and performance we also investigate the issue of optimality and present an improved least square formulation of the problem this approach leads to a separable three pass implementation of a rotation using one dimensional convolutions only we provide explicit filter formulas for several continuous signal models including spline and bandlimited representations finally we present rotation experiments and compare the currently standard techniques with the various versions of our algorithm our results indicate that the present algorithm in its higher order versions outperforms all standard high accuracy methods of which we are aware both in terms of speed and quality its computational complexity increases linearly with the order of accuracy the best quality results are obtained with the sine based algorithm which can be implemented using simple one dimensional ffts

a penalization method to take into account obstacles in incompressible viscous flows from the navier stokes brinkman model a penalization method has been derived by several authors to compute incompressible navier stokes equations around obstacles in this paper convergence theorems and error estimates are derived for two kinds of penalization the first one corresponds to l 2 penalization inducing a darcy equation in the solid body the second one corresponds to a h 1 penalization and induces a brinkman equation in the body numerical tests are performed to confirm the efficiency and accuracy of the method

uniqueness of strongly regular graphs having minimal p rank abstract let be a strongly regular graph with adjacency matrix a let i be the identity matrix and let p be a prime we study the p rank of the matrices a ci for integral c and want to characterize among all strongly regular graphs with a given parameter set those for which a p rank is minimal

a fast finite volume method for conservative space fractional diffusion equations in convex domains we develop a fast finite volume method for variable coefficient conservative space fractional diffusion equations in convex domains via a volume penalization approach the method has an optimal storage and an almost linear computational complexity the method retains second order accuracy without requiring a richardson extrapolation numerical results are presented to show the utility of the method

hybrid control for switched linear systems with average dwell time this technical note presents a hybrid control scheme for the output feedback control of switched linear systems with average dwell time the proposed hybrid controller consists of a standard switching output feedback control law and a supervisor enforcing a reset rule for the switching controller states at each switching instant this hybrid control scheme provides an efficient and systema tic way for designing average dwell time switched linear control systems in the sense that the boundary condition can be incorporated into the synthesis problem in a convex formulation specifically bo th full order and reduced order controllers with guaranteed stability and optimal weighted h1 performance will be solved by linear matrix inequality lmi optimizations simulation studies are included to illustrate the effectiveness of the proposed approach index terms switching control average dwell time controller state reset full order and reduced order controllers lmi

a fast computation technique for the direct numerical simulation of rigid particulate flows in this paper we present a computation technique for the direct numerical simulation of freely moving rigid bodies in fluids we solve three dimensional laminar flow problems using a control volume approach the key feature of this approach is that the computational overhead relative to a pure fluid solver to solve for the motion of rigid particle is very small the formulation is convenient for handling irregular geometries we present results for the sedimentation of particles of different shapes convergence tests are presented to assess the order of accuracy of the numerical scheme various test cases are considered and the numerical results are compared with experimental values to validate the code due to the ability to perform fast computations this method has been used for animations and its application to the direct numerical simulation of turbulent particulate flows merits investigation the technique is not restricted to any constitutive model of the suspending fluid hence it may potentially be used in large eddy simulations les or reynolds averaged navier stokes rans type simulations

on the kalman yakubovich popov lemma the purpose of this note is to present a new elementary proof for the multivariable k y p lemma a minimum of linear algebra and finite dimensional convexity theory is used

finite frequency memory output feedback controller design for t s fuzzy dynamical systems in this paper we will investigate the problem of finite frequency memory fixed order output feedback controller design for takagi sugeno t s fuzzy affine systems it is assumed that the disturbances reside in a finite frequency range i e the low middle or high frequency range the objective is to design a memory piecewise affine pwa controller by using past output measurements to guarantee the asymptotic stability of the resulting closed loop system with a prescribed finite frequency mathscr h infty performance via the system state input augmentation a novel descriptor system approach is proposed to facilitate the controller design all the design conditions are formulated in the form of linear matrix inequalities it is also proven that the mathscr h infty performance can be improved with the memory control strategy finally simulation studies are presented to show the effectiveness of the proposed design method

stability analysis with dissipation inequalities and integral quadratic constraints this technical note considers the stability of a feedback connection of a known linear time invariant system and a perturbation the input output behavior of the perturbation is described by an integral quadratic constraint iqc iqc stability theorems can be formulated in the frequency domain or with a time domain dissipation inequality the two approaches are connected by a non unique factorization of the frequency domain iqc multiplier the factorization must satisfy two properties for the dissipation inequality to be valid first the factorization must ensure the time domain iqc holds for all finite times second the factorization must ensure that a related matrix inequality when feasible has a positive semidefinite solution this technical note shows that a class of frequency domain iqc multipliers has a factorization satisfying these two properties thus the dissipation inequality test with an appropriate factorization can be used with no additional conservatism

brief paper factorization of multipliers in passivity and iqc analysis multipliers are often used to find conditions for the absolute stability of lur e systems they can be used either in conjunction with passivity theory or within the more recent framework of integral quadratic constraints iqcs we compare the use of multipliers in both approaches passivity theory requires that the multipliers have a canonical factorization and it has been suggested in the literature that this represents an advantage of the iqc theory we consider sufficient conditions on the nonlinearity class for the associated multipliers to have a canonical factorization

a fourier penalty method for solving the time dependent maxwell s equations in domains with curved boundaries we present a high order fourier penalty method for the maxwell s equations in the vicinity of perfect electric conductor boundary conditions the approach relies on extending the smooth non periodic domain of the equations to a periodic domain by removing the exact boundary conditions and introducing an analytic forcing term in the extended domain the forcing or penalty term is chosen to systematically enforce the boundary conditions to high order in the penalty parameter which then allows for higher order numerical methods we present an efficient numerical method for constructing the penalty term and discretize the resulting equations using a fourier spectral method we demonstrate convergence orders of up to 3 5 for the one dimensional maxwell s equations and show that the numerical method does not suffer from dispersion or pollution errors we also illustrate the approach in two dimensions and demonstrate convergence orders of 2 5 for transverse magnetic modes and 1 5 for the transverse electric modes we conclude the paper with numerous test cases in dimensions two and three including waves traveling in a bent waveguide and scattering off of a windmill like geometry

interlacing eigenvalues and graphs abstract we give several old and some new applications of eigenvalue interlacing to matrices associated to graphs bounds are obtained for characteristic numbers of graphs such as the size of a maximal co clique the chromatic number the diameter and the bandwidth in terms of the eigenvalues of the standard adjacency matrix or the laplacian matrix we also deal with inequalities and regularity results concerning the structure of graphs and block designs

robustness with dynamic iqcs an exact state space characterization of nominal stability with applications to robust estimation for robustness analysis with integral quadratic constraints we formulate a new positivity condition on the solution of the corresponding linear matrix inequality which is necessary and sufficient for nominal stability of the underlying system the application of this technical result is illustrated by a complete solution of the l 2 gain and robust h 2 estimator design problems if the uncertainties are characterized by dynamic integral quadratic constraints

sparse low energy decomposition for viscous conservation laws for viscous conservation laws solutions contain smooth but high contrast features which require the use of fine grids to properly resolve on coarse grids these high contrast jumps resemble shocks rather than their true viscous profiles which could lead to issues in the numerical approximation of their underlying dynamics in many cases the equations of motion emit traveling wave solutions which can be used to represent the viscous profiles analytically the traveling wave solutions can be thought of as a lower dimensional representation of the motion since they contain information from the evolution equation but are constant along certain time space curves using a parameterized basis involving the traveling waves along with the sparse low energy decompositions found in imaging sciences we propose an approximation to viscous conservation laws which separates the coarse smooth component from the sharp fine one our method provides an appropriate approximation to the solution on a coarse grid thereby accurately under resolving the viscous profile this is similar to the philosophy of shock capturing methods in the sense that we want to capture the viscous front without needing to resolve the profile theoretical results on the consistency of our method are shown in general we provide several computational examples for convex and non convex fluxes

a fictitious domain approach to the direct numerical simulation of incompressible viscous flow past moving rigid bodies application to particulate flow abstract in this article we discuss a methodology that allows the direct numerical simulation of incompressible viscous fluid flow past moving rigid bodies the simulation methods rest essentially on the combination of a lagrange multiplier based fictitious domain methods which allow the fluid flow computations to be done in a fixed flow region b finite element approximations of the navier stokes equations occurring in the global model c time discretizations by operator splitting schemes in order to treat optimally the various operators present in the model the above methodology is particularly well suited to the direct numerical simulation of particulate flow such as the flow of mixtures of rigid solid particles and incompressible viscous fluids possibly non newtonian we conclude this article with the presentation of the results of various numerical experiments including the simulation of store separation for rigid airfoils and of sedimentation and fluidization phenomena in two and three dimensions

a fourier spectral method for the navier stokes equations with volume penalization for moving solid obstacles an efficient numerical scheme to compute flows past rigid solid bodies moving through viscous incompressible fluid is presented solid obstacles of arbitrary shape are taken into account using the volume penalization method to impose no slip boundary condition the 2d navier stokes equations written in the vorticity streamfunction formulation are discretized using a fourier pseudo spectral scheme four different time discretization schemes of the penalization term are proposed and compared the originality of the present work lies in the implementation of time dependent penalization which makes the above method capable of solving problems where the obstacle follows an arbitrary motion fluid solid coupling for freely falling bodies is also implemented the numerical method is validated for different test cases the flow past a cylinder couette flow between rotating cylinders sedimentation of a cylinder and a falling leaf with elliptical shape

binary codes derived from the hoffman singleton and higman sims graphs some binary linear codes of length 50 and 100 are constructed using the adjacency matrices of the hoffman singleton graph and the higman sims graph some of the codes are optimal or nearly optimal for the given length and dimension the dual codes admit majority logic decoding

exact memory and memoryless control of linear systems with time varying input delay using dynamic iqcs input delay is an important type of actuator nonlinearity in control systems in this paper we will address the output feedback control synthesis problem for linear systems with time varying input delay under the integral quadratic constraint iqc framework a new exact memory control scheme is first proposed which consists of a standard linear output feedback control law and an internal delay loop the delay loop is embedded in the controller structure so as to reproduce the input delay behavior of the plant by using quadratic lyapunov functions incorporated with dynamic iqc multipliers the resulting dynamic output feedback delay control synthesis problem is fully characterized by a set of linear matrix inequalities lmis which are convex on all design variables including the scaling factors associated with the iqc multipliers moreover the corresponding result on memoryless control is also derived for the case when the plant input delay information is not available for feedback control an application to network systems has been used to illustrate the effectiveness and usefulness of the proposed approach

a volume penalization method for incompressible flows and scalar advection diffusion with moving obstacles a volume penalization method for imposing homogeneous neumann boundary conditions in advection diffusion equations is presented thus complex geometries which even may vary in time can be treated efficiently using discretizations on a cartesian grid a mathematical analysis of the method is conducted first for the one dimensional heat equation which yields estimates of the penalization error the results are then confirmed numerically in one and two space dimensions simulations of two dimensional incompressible flows with passive scalars using a classical fourier pseudo spectral method validate the approach for moving obstacles the potential of the method for real world applications is illustrated by simulating a simplified dynamical mixer where for the fluid flow and the scalar transport no slip and no flux boundary conditions are imposed respectively

stability analysis of systems with uncertain time varying delays stability of systems in the presence of bounded uncertain time varying delays in the feedback loop is studied the delay parameter is assumed to be an unknown time varying function for which the upper bounds on the magnitude and the variation are given the stability problem is treated in the integral quadratic constraint iqc framework criteria for verifying robust stability are formulated as feasibility problems over a set of frequency dependent linear matrix inequalities the criteria can be equivalently formulated as semi definite programs sdp using kalman yakubovich popov lemma as such checking robust stability can be performed in a computationally efficient fashion

synthesis of non rational controllers for linear delay systems this paper addresses the control of linear delay systems using non rational controllers the structure of the controller is chosen so as to copy the structure of the plant reproducing the delays in the state and in the output the resulting stabilization and performance design problems are entirely expressed as linear matrix inequalities although the design inequalities are based on delay independent stability conditions the overall design is delay dependent in the sense that the controller makes use of the delay parameter of the plant this parameter is assumed to be constant yet arbitrary using non rational controllers we overcome the main difficulty faced when designing rational controllers for linear delay systems which is to incorporate in the design problem the matrix multiplier used to prove stability with respect to the delayed part of the system we illustrate the paper with several examples and provide extensive comparisons with existent results

nineteen dubious ways to compute the exponential of a matrix twenty five years later in principle the exponential of a matrix could be computed in many ways methods involving approximation theory differential equations the matrix eigenvalues and the matrix characteristic polyn

flusi a novel parallel simulation tool for flapping insect flight using a fourier method with volume penalization we introduce flusi a fully parallel open source software package for pseudospectral simulations of three dimensional flapping flight in viscous flows it is freely available for noncommercial use from github https github com pseudospectators flusi the computational framework runs on high performance computers with distributed memory architectures the discretization of the three dimensional incompressible navier stokes equations is based on a fourier pseudospectral method with adaptive time stepping the complex time varying geometry of insects with rigid flapping wings is handled using the volume penalization method the modules characterizing the insect geometry flight mechanics and wing kinematics are described validation tests for different benchmarks illustrate the efficiency and precision of the approach finally computations for a model insect in the turbulent regime demonstrate the versatility of the software

integral quadratic constraint approach vs multiplier approach integral quadratic constraints iqc arise in many optimal and or robust control problems the iqc approach can be viewed as a generalization of the classical multiplier approach in the absolute stability theory in this paper we study the relationship between the two approaches for robust stability analysis using a slightly modified multiplier approach we show that the existence of an iqc is equivalent to the existence of a multiplier in most known cases it is hoped that this result provides some new insight into both approaches and makes them more useful in robust control applications

fourier spectral and wavelet solvers for the incompressible navier stokes equations with volume penalization convergence of a dipole wall collision in this study we use volume penalization to mimic the presence of obstacles in a flow or a domain with no slip boundaries this allows in principle the use of fast fourier spectral methods and coherent vortex simulation techniques based on wavelet decomposition of the flow variables to compute turbulent wall bounded flow or flows around solid obstacles by simply adding one term in the equation convergence checks are reported using a recently revived and unexpectedly difficult dipole wall collision as a benchmark computation several quantities like the vorticity isolines truncation error kinetic energy and enstrophy are inspected for a collision of a dipole with a no slip wall and compared with available benchmark data obtained with a standard chebyshev pseudospectral method we quantify the possible deteriorating effects of the gibbs phenomenon present in the fourier based schemes due to continuity restrictions of the penalized navier stokes equations on the wall it is found that gibbs oscillations have a negligible effect on the flow evolution allowing higher order recovery of the accuracy on a fourier basis by means of postprocessing an advantage of coherent vortex simulations on the other hand is that the degrees of freedom of the flow computation can strongly be reduced in this study we quantify the possible reduction of degrees of freedom while keeping the accuracy for an optimal convergence scenario the penalization parameter has to scale with the number of fourier and wavelet modes in addition an implicit treatment of the darcy drag term in the penalized navier stokes equations is beneficial since this allows one to set the time step independent from the penalization parameter without additional computational or memory requirements

error correction for index coding with side information a problem of index coding with side information was first considered by birk and kol in 1998 in this study a generalization of index coding scheme where transmitted symbols are subject to errors is studied error correcting methods for such a scheme and their parameters are investigated in particular the following question is discussed given the side information hypergraph of index coding scheme and the maximal number of erroneous symbols what is the shortest length of a linear index code such that every receiver is able to recover the required information this question turns out to be a generalization of the problem of finding a shortest length error correcting code with a prescribed error correcting capability in the classical coding theory the singleton bound and two other bounds referred to as the bound and the bound for the optimal length of a linear error correcting index code ecic are established for large alphabets a construction based on concatenation of an optimal index code with a maximum distance separable classical code is shown to attain the singleton bound for smaller alphabets however this construction may not be optimal a random construction is also analyzed it yields another inexplicit bound on the length of an optimal linear ecic further the problem of error correcting decoding by a linear ecic is studied it is shown that in order to decode correctly the desired symbol the decoder is required to find one of the vectors belonging to an affine space containing the actual error vector the syndrome decoding is shown to produce the correct output if the weight of the error pattern is less or equal to the error correcting capability of the corresponding ecic finally the notion of static ecic which is suitable for use with a family of instances of an index coding problem is introduced several bounds on the length of static ecics are derived and constructions for static ecics are discussed connections of these codes to weakly resilient boolean functions are established

integral quadratic constraints for delayed nonlinear and parameter varying systems the stability and performance of nonlinear and linear parameter varying lpv time delayed systems are analyzed first the input output behavior of the time delay operator is bounded in the frequency domain by integral quadratic constraints iqcs a simple geometric interpretation is used to derive new iqcs for both constant and varying delays second the performance of nonlinear and lpv delayed systems is bounded using dissipation inequalities that incorporate iqcs the nonlinear or lpv part of the system is treated directly in the analysis and not bounded by iqcs this step makes use of recent results that show under mild technical conditions that an iqc has an equivalent representation as a finite horizon time domain constraint a numerical example with a nonlinear delayed system is provided to demonstrate the effectiveness of the method

a versatile sharp interface immersed boundary method for incompressible flows with complex boundaries a sharp interface immersed boundary method for simulating incompressible viscous flow past three dimensional immersed bodies is described the method employs a multi dimensional ghost cell methodology to satisfy the boundary conditions on the immersed boundary and the method is designed to handle highly complex three dimensional stationary moving and or deforming bodies the complex immersed surfaces are represented by grids consisting of unstructured triangular elements while the flow is computed on non uniform cartesian grids the paper describes the salient features of the methodology with special emphasis on the immersed boundary treatment for stationary and moving boundaries simulations of a number of canonical two and three dimensional flows are used to verify the accuracy and fidelity of the solver over a range of reynolds numbers flow past suddenly accelerated bodies are used to validate the solver for moving boundary problems finally two cases inspired from biology with highly complex three dimensional bodies are simulated in order to demonstrate the versatility of the method

on stability of discrete time lti systems with varying time delays this technical note concerns robust stability analysis of discrete time linear time invariant lti systems with varying time delays the stability problem is tackled under the integral quadratic constraint iqc framework the time delay system in question is viewed as feedback interconnection of an lti stable operator and the so called delay difference operator for which iqc characterizations are required in order to apply iqc analysis one of the main contributions of the technical note is to derive novel iqc characterizations for the delay difference operator based on the assumption that the time varying delay parameter and its variation are bounded a set of new stability criteria emerges as the result numerical examples are given to illustrate the effectiveness of the new stability criteria another distinct contribution is the idea of introducing bounds on the so called m step total variation and m step average variation of the time varying delay parameter which is proven to be useful in stability analysis

approximation of the laplace and stokes operators with dirichlet boundary conditions through volume penalization a spectral viewpoint we report the results of a study on the spectral properties of laplace and stokes operators modified with a volume penalization term designed to approximate dirichlet conditions in the limit when a penalization parameter eta tends to zero the eigenvalues and eigenfunctions are determined either analytically or numerically as functions of eta both in the continuous case and after applying fourier or finite difference discretization schemes for fixed eta we find that only the part of the spectrum corresponding to eigenvalues lambda lesssim eta 1 1 approaches dirichlet boundary conditions while the remainder of the spectrum is made of uncontrolled spurious wall modes the penalization error for the controlled eigenfunctions is estimated as a function of eta and lambda surprisingly in the stokes case we show that the eigenfunctions approximately satisfy with a precision o eta o navier slip boundary conditions with slip length equal to sqrt eta moreover for a given discretization we show that there exists a value of eta corresponding to a balance between penalization and discretization errors below which no further gain in precision is achieved these results shed light on the behavior of volume penalization schemes when solving the navier stokes equations outline the limitations of the method and give indications on how to choose the penalization parameter in practical cases