cutfem topology optimization of 3d laminar incompressible flow problems abstract this paper studies the characteristics and applicability of the cutfem approach of burman et al 2014 as the core of a robust topology optimization framework for 3d laminar incompressible flow and species transport problems at low reynolds numbers r e 200 cutfem is a methodology for discretizing partial differential equations on complex geometries by immersed boundary techniques in this study the geometry of the fluid domain is described by an explicit level set method where the parameters of a discretized level set function are defined as functions of the optimization variables the fluid behavior is modeled by the incompressible navier stokes equations species transport is modeled by an advection diffusion equation the governing equations are discretized in space by a generalized extended finite element method face oriented ghost penalty terms are added for stability reasons and to improve the conditioning of the system the boundary conditions are enforced weakly via nitsche s method the emergence of isolated volumes of fluid surrounded by solid during the optimization process leads to a singular analysis problem an auxiliary indicator field is modeled to identify these volumes and to impose a constraint on the average pressure numerical results for 3d steady state and transient problems demonstrate that the cutfem analyses are sufficiently accurate and the optimized designs agree well with results from prior studies solved in 2d or by density approaches

numerical analysis of a finite element volume penalty method we present here some contributions to the numerical analysis of the penalty method in the finite element context we are especially interested in the ability provided by this approach to use cartesian non boundary fitted meshes to solve elliptic problems in complicated domain in the spirit of fictitious domains the initial problem is replaced by a penalized one posed over a simply shaped domain which covers the original one this method relies on two parameters namely h space discretization parameter and varepsilon penalty parameter we propose here a general strategy to estimate the error in both parameters and we present how it can be applied to various situations we pay special attention to a scalar version of the rigid motion constraint for fluid particle flows

numerical method for hydrodynamic transport of inhomogeneous polymer melts we introduce a mesoscale method for simulating hydrodynamic transport and self assembly of inhomogeneous polymer melts in pressure driven and drag induced flows this method extends dynamic self consistent field theory dscft into the hydrodynamic regime where bulk material transport and viscoelastic effects play a significant role the method combines four distinct components as a single coupled system including 1 non equilibrium self consistent field theory describing block copolymer self assembly 2 multi fluid navier stokes type hydrodynamics for tracking material transport 3 constitutive equations modeling viscoelastic phase separation and 4 rigid wall fields which represent moving channel boundaries machine components and nano particulate fillers we also present an efficient pseudo spectral implementation for this set of coupled equations which enables practical application of the model in periodic domains we validate the model by reproducing well known phenomena including equilibrium diblock meso phases analytic stokes flows and viscoelastic phase separation of glassy elastic polymer melts we also demonstrate the stability and accuracy of the numerical implementation by examining its convergence under grid size refinement

iterative brinkman penalization for remeshed vortex methods we introduce an iterative brinkman penalization method for the enforcement of the no slip boundary condition in remeshed vortex methods in the proposed method the brinkman penalization is applied iteratively only in the neighborhood of the body this allows for using significantly larger time steps than what is customary in the brinkman penalization thus reducing its computational cost while maintaining the capability of the method to handle complex geometries we demonstrate the accuracy of our method by considering challenging benchmark problems such as flow past an impulsively started cylinder and normal to an impulsively started and accelerated flat plate we find that the present method enhances significantly the accuracy of the brinkman penalization technique for the simulations of highly unsteady flows past complex geometries

noise tolerant parallel learning of geometric concepts we present several efficient parallel algorithms for pac learning geometric concepts in a constant dimensional space the algorithms are robust even against malicious classification noise of any rate less than 1 2 we first give an efficient noise tolerant parallel algorithm to pac learn the class of geometric concepts defined by a polynomial number of d 1 dimensional hyperplanes against an arbitrary distribution where each hyperplane has a slope from a set of known slopes we then describe how boosting techniques can be used so that our algorithms dependence on and does not depend ond next we give an efficient noise tolerant parallel algorithm to pac learn the class of geometric concepts defined by a polynomial number of d 1 dimensional hyperplanes of unrestricted slopes against a uniform distribution we then show how to extend our algorithm to learn this class against any unknown product distribution next we define a complexity measure of any setsof d 1 dimensional surfaces that we call thevariantofsand prove that the class of geometric concepts defined by surfaces of polynomial variant can be efficiently learned in parallel under a product distribution even under malicious classification noise furthermore we show that the vc dimension of the class of geometric concepts defined by a single surface of variant one is finally we give an efficient parallel noise tolerant algorithm to pac learn any 2 dimensional geometric concept defined by a setsof 1 dimensional surfaces of polynomial length under a uniform distribution

modeling a no slip flow boundary with an external force field abstract a novel technique related to peskin s immersed boundary approach is used to introduce solid surfaces into a simulated flow field the navier stokes equations permit the presence of an externally imposed body force that may vary in space and time forces are chosen to lie along a desired surface and to have a magnitude and direction opposing the local flow such that the flow is brought to rest on an element of the surface for unsteady viscous flow the direct calculation of the needed force is facilitated by a feedback scheme in which the velocity is used to iteratively determine the desired value in particular we determine the surface body force from the relation f x 5 t t 0 u x 5 t dt u x 5 t for surface points x 5 velocity u time t and negative constants and examples are presented which include 2d flow around cylinders 3d turbulent channel flow where one boundary is simulated with a force field and turbulent channel flow over a riblet covered surface while the new method may be applied to complex geometries on a non cartesian mesh we have chosen to use a simple cartesian grid all simulations are done with a spectral code in a single computational domain without any mapping of the mesh

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

optimal design for non newtonian flows using a topology optimization approach we study non newtonian effects on the layout and geometry of flow channels using a material distribution based topology optimization approach the flow is modeled with the single relaxation hydrodynamic lattice boltzmann method and the shear dependence of viscosity is included through the carreau yasuda model for non newtonian fluids to represent the viscosity of blood in this model we use non newtonian similarity further we introduce a scaling to decrease the effects of the non newtonian model in porous regions in order to stabilize the coupling of the lbm porosity and non newtonian flow models for the resulting flow model we derive the non newtonian sensitivity analysis for steady state conditions and illustrate the non newtonian effect on channel layouts for a 2d dual pipe design problem at different reynolds numbers

smooth optimization with approximate gradient we show that the optimal complexity of nesterov s smooth first order optimization algorithm is preserved when the gradient is computed only up to a small uniformly bounded error in applications of this method to semidefinite programs this means in some instances computing only a few leading eigenvalues of the current iterate instead of a full matrix exponential which significantly reduces the method s computational cost this also allows sparse problems to be solved efficiently using sparse maximum eigenvalue packages

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

transitions between streamline topologies of structurally stable hamiltonian flows in multiply connected domains abstract we consider hamiltonian vector fields with a dipole singularity satisfying the slip boundary condition in two dimensional multiply connected domains one example of such hamiltonian vector fields is an incompressible and inviscid flow in exterior multiply connected domains with a uniform flow whose hamiltonian is called the stream function here we are concerned with topological structures of the level sets of the hamiltonian which we call streamlines by analogy from incompressible fluid flows classification of structurally stable streamline patterns has been considered in yokoyama and sakajo 2013 where a procedure to assign a unique sequence of words called the maximal word to these patterns is proposed thanks to this procedure we can identify every streamline pattern with its representing sequence of words up to topological equivalence in the present paper based on the theory of word representations we propose a combinatorial method to provide a list of possible transient structurally unstable streamline patterns between two different structurally stable patterns by simply comparing their maximal word representations without specifying any hamiltonian although this method cannot deal with topological streamline changes induced by bifurcations it reveals the existence of many non trivial global transitions in a generic sense we also demonstrate how the present theory is applied to fluid flow problems with vortex structures

computation of three dimensional multiphase flow dynamics by fully coupled immersed flow fcif solver abstract this work presents a fully coupled immersed flow fcif solver for the three dimensional simulation of fluid fluid interaction by coupling two distinct flow solvers using an immersed boundary ib method the fcif solver captures dynamic interactions between two fluids with disparate flow properties while retaining the desirable simplicity of non boundary conforming grids for illustration we couple an ib based unsteady reynolds averaged navier stokes urans simulator with a depth integrated long wave solver for the application of slug development with turbulent gas and laminar liquid we perform a series of validations including turbulent laminar flows over prescribed wavy boundaries and freely evolving viscous fluids these confirm the effectiveness and accuracy of both one way and two way coupling in the fcif solver finally we present a simulation example of the evolution from a stratified turbulent laminar flow through the initiation of a slug that nearly bridges the channel the results show both the interfacial wave dynamics excited by the turbulent gas forcing and the influence of the liquid on the gas turbulence these results demonstrate that the fcif solver effectively captures the essential physics of gas liquid interaction and can serve as a useful tool for the mechanistic study of slug generation in two phase gas liquid flows in channels and pipes

convergence analysis of a penalization method for the three dimensional motion of a rigid body in an incompressible viscous fluid we present and analyze a penalization method which extends the the method of ph angot c h bruneau and p fabrie numer math 81 1999 pp 487 520 to the case of a rigid body moving freely in an incompressible fluid the fluid solid system is viewed as a single variable density flow the interface is captured by a color function satisfying a transport equation the solid velocity is computed by averaging at every time the flow velocity in the solid phase this velocity is used to penalize the flow velocity at the fluid solid interface and to move the interface numerical illustrations are provided to illustrate our convergence result a discussion of our result in the light of existing existence results is also given

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

excessive gap technique in nonsmooth convex minimization in this paper we introduce a new primal dual technique for convergence analysis of gradient schemes for nonsmooth convex optimization as an example of its application we derive a primal dual gradient method for a special class of structured nonsmooth optimization problems which ensures a rate of convergence of order o 1 k where k is the iteration count another example is a gradient scheme which minimizes a nonsmooth strongly convex function with known structure with rate of convergence o 1 k2 in both cases the efficiency of the methods is higher than the corresponding black box lower complexity bounds by an order of magnitude

parallel linear programming in fixed dimension almost surely in constant time for any fixed dimension d thelinear programming problem with n inequality constraints can be solved on a probabilistic crcw pram with o n processors almost surely in constant time the algorithm always findsthe correct solution with nd log 2 d processors the probability that the algorithm will not finish within o d 2 log 2 d time tends to zero exponentially with n authors abstract

universal meshes a method for triangulating planar curved domains immersed in nonconforming meshes summary we introduce a new method to triangulate planar curved domains that transforms a specific collection of triangles in a background mesh to conform to the boundary in the process no new vertices are introduced and connectivities of triangles are left unaltered the method relies on a novel way of parameterizing an immersed boundary over a collection of nearby edges with its closest point projection to guarantee its robustness we require that the domain be c2 regular the background mesh be sufficiently refined near the boundary and that specific angles in triangles near the boundary be strictly acute the method can render both straight edged and curvilinear triangulations for the immersed domain the latter includes curved triangles that conform exactly to the immersed boundary and ones constructed with isoparametric mappings to interpolate the boundary at select points high order finite elements constructed over these curved triangles achieve optimal accuracy which has customarily proven difficult in numerical schemes that adopt nonconforming meshes aside from serving as a quick and simple tool for meshing planar curved domains with complex shapes the method provides significant advantages for simulating problems with moving boundaries and in numerical schemes that require iterating over the geometry of domains with no conformity requirements the same background mesh can be adopted to triangulate a large family of domains immersed in it including ones realized over several updates during the coarse of simulating problems with moving boundaries we term such a background mesh as a universal mesh for the family of domains it can be used to triangulate universal meshes hence facilitate a framework for finite element calculations over evolving domains while using only fixed background meshes furthermore because the evolving geometry can be approximated with any desired order numerical solutions can be computed with high order accuracy we present demonstrative examples using universal meshes to simulate the interaction of rigid bodies with stokesian fluids copyright 2014 john wiley sons ltd

penalization modeling of a limiter in the tokamak edge plasma an original penalization method is applied to model the interaction of magnetically confined plasma with limiter in the frame of a minimal transport model for ionic density and parallel momentum the limiter is considered as a pure particle sink for the plasma and consequently the density and the momentum are enforced to be zero inside comparisons of the numerical results with one dimensional analytical solutions show a very good agreement in particular the penalization scheme followed in this paper tends to ensure an almost sonic plasma condition at the plasma obstacle interface bohm like criterion with relatively weak dependence on the target mach number profile within the obstacle the new system being solved in a periodic obstacle free domain an efficient pseudo spectral algorithm based on a fast fourier transform is also proposed and associated with an exponential filtering of the unphysical oscillations due to gibbs phenomenon finally the efficiency of the method is illustrated by investigating the flow spreading from the plasma core to the scrape off layer at the wall in a two dimensional system with one then two neighboring limiters

glass transition seen through asymptotic expansions soft glassy materials exhibit the so called glassy transition which means that the behavior of the model at a low shear rate changes when a certain parameter which we call the glass parameter crosses a critical value this behavior goes from a newtonian behavior to a herschel bulkley behavior through a power law type behavior at the transition point in a previous paper we rigorously proved that the hebraud lequeux model a fokker planck like description of soft glassy material exhibits such a glass transition but the method we used was very specific to the one dimensional setting of the model and as a preparation for generalizing this model to take into account multidimensional situations we look for another technique to study the glass transition of this type of model in this paper we shall use matched asymptotic expansions for such a study the difficulties encountered when using asymptotic expansions for the hebraud lequeux model are that multiple ansaetze have to be used even though the init

on a free boundary problem abstract in this paper we will be concerned with the existence of positive solutions of the following discontinuous problem p u f u h u in u h on where is the unit ball of r n n 3 and h is the heaviside function the problem is written in an equivalent integral equation form the shape of the free boundary problem x u x is studied for when is a positive real parameter f and h are given functions

a finite difference method for the variable coefficient poisson equation on hierarchical cartesian meshes we consider problems governed by a linear elliptic equation with varying coefficients across internal interfaces the solution and its normal derivative can undergo significant variations through these internal boundaries we present a compact finite difference scheme on a tree based adaptive grid that can be efficiently solved using a natively parallel data structure the main idea is to optimize the truncation error of the discretization scheme as a function of the local grid configuration to achieve second order accuracy numerical illustrations are presented in two and three dimensional configurations finite difference method hierarchical cartesian grid octree quadtree variable coefficient poisson equation 1

variational methods for non differentiable functionals and their applications to partial differential equations the interest in extending the theory of nonlinear partial differential equations pde to pde with discontinuous nonlinearities dnde has been increasing daily the reasons are twofold 1 many free boundary problems may be reduced to boundary value problems of dnde cf chang 16 71 chang and jiang 9 and fleishman and mahar 121 2 sometimes in dealing with a pde problem it is beneficial to put the original pde into a large category for instance dnde cf chang 7 simply speaking a dnde is as follows

numerical simulation of water flow around a rigid fishing net this paper is devoted to the simulation of the flow around and inside a rigid axisymmetric net we describe first how experimental data have been obtained we show in detail the modelization the model is based on a reynolds averaged navier stokes turbulence model penalized by a term based on the brinkman law at the out boundary of the computational box we have used a ghost boundary condition we show that the corresponding variational problem has a solution then the numerical scheme is given and the paper finishes with numerical simulations compared with the experimental data

learnability and the vapnik chervonenkis dimension valiant s learnability model is extended to learning classes of concepts defined by regions in euclidean space e n the methods in this paper lead to a unified treatment of some of valiant s results along with previous results on distribution free convergence of certain pattern recognition algorithms it is shown that the essential condition for distribution free learnability is finiteness of the vapnik chervonenkis dimension a simple combinatorial parameter of the class of concepts to be learned using this parameter the complexity and closure properties of learnable classes are analyzed and the necessary and sufficient conditions are provided for feasible learnability

an immersed boundary method using unstructured anisotropic mesh adaptation combined with level sets and penalization techniques the interest on embedded boundary methods increases in computational fluid dynamics cfd because they simplify the mesh generation problem in the case of the navier stokes equations the same simplifications occur for the simulation of multi physics flows the coupling of fluid solid interactions in situation of large motions or deformations to give a few examples nevertheless an accurate treatment of the wall boundary conditions remains an issue of the method in this work the wall boundary conditions are easily taken into account through a penalization technique and the accuracy of the method is recovered using mesh adaptation thanks to the potential of unstructured meshes several classical examples are used to demonstrate that claim

a multigrid ghost point level set method for incompressible navier stokes equations on moving domains with curved boundaries abstract in this paper we present a numerical approach to solve the navier stokes equations on moving domains with second order accuracy the space discretization is based on the ghost point method which falls under the category of unfitted boundary methods since the mesh does not adapt to the moving boundary the equations are advanced in time by using crank nicholson the momentum and continuity equations are solved simultaneously for the velocity and the pressure by adopting a proper multigrid approach to avoid the checkerboard instability for the pressure a staggered grid is adopted where velocities are defined at the sides of the cell and the pressure is defined at the centre the lack of uniqueness for the pressure is circumvented by the inclusion of an additional scalar unknown representing the average divergence of the velocity and an additional equation to set the average pressure to zero several tests are performed to simulate the motion of an incompressible fluid around a moving object as well as the lid driven cavity tests around steady objects the object is implicitly defined by a level set approach that allows a natural computation of geometrical properties such as distance from the boundary normal directions and curvature different shapes are tested circle ellipse and flower numerical results show the second order accuracy for the velocity and the divergence that decays to zero with second order and the efficiency of the multigrid that is comparable with the tests available in literature for rectangular domains without objects showing that the presence of a complex shaped object does not degrade the performance

on the generalization of the hebraud lequeux model to multidimensional flows in this article we build a model for multidimensional flows in soft glassy materials the construction of the model is based on the ideas of hebraud and lequeux but care is taken to build a frame indifferent multidimensional model the main goal of this article is to prove that the methodology we have developed to study the well posedness and the glass transition for the original hebraud lequeux model can be successfully generalized thus this work may be used as a starting point for more sophisticated studies in the modeling of general flows of glassy materials

analysis of the fully discrete fat boundary method the fat boundary method is a method of the fictitious domain class which was proposed to solve elliptic problems in complex geometries with non conforming meshes it has been designed to recover optimal convergence at any order despite of the non conformity of the mesh and without any change in the discrete laplace operator on the simple shape domain we propose here a detailed proof of this high order convergence and propose some numerical tests to illustrate the actual behaviour of the method

an adaptive residual based splitting approach for the penalized navier stokes equations the interest on immersed boundary methods ibm is increasing in computational fluid dynamics as they simplify the mesh generation problem in this work we consider an approach based on the addition of a penalty term to the navier stokes equations to account for the wall boundary conditions to discretize the resulting equations we use a residual distribution approach previously developed by some of the authors to adapt the method to the ibm considered we developed a new formulation of residual distribution based on a strang splitting method in time coupling an implicit asymptotic integration procedure of the penalization ode with a simplified explicit residual distribution for the navier stokes equations the first method provides an operator which is exact up to orders 2 with the penalty parameter assuming values of the order of 10 10 a modification of the solution gradient reconstruction necessary for the evaluation of the viscous fluxes is also introduced in the paper this guarantees that correct physical values of the viscous stresses are recovered in vicinity of the solid we show formally and numerically that the approach proposed is second order accurate for smooth solutions we evaluate its potential for ibm by coupling the resulting method with unstructured mesh adaptation on wall boundaries several steady and time dependent tests are used to show the promising features of the method proposed

modeling and simulation of fish like swimming modeling and simulation of two dimensional flows past deformable bodies are considered the incompressible navier stokes equations are discretized in space onto a fixed cartesian mesh and the displacement of deformable objects through the fluid is taken into account using a penalization method the interface between the solid and the fluid is tracked using a level set description so that it is possible to simulate several bodies freely evolving in the fluid as an illustration of the methods fish like locomotion is analyzed in terms of propulsion efficiency underwater maneuvering and school swimming are also explored

numerical simulation of a cylinder in uniform flow in this study a virtual boundary technique is applied to the numerical simulation of stationary and moving cylinders in uniform flow this approach readily allows the imposition of a no slip boundary within the flow field by a feedback forcing term added to the momentum equations in the present work this technique is used with a high order finite difference method effectively eliminating spurious oscillations caused by the feedback forcing when used with spectrally discretized flow solvers very good agreement is found between the present calculations and previous computational and experimental results for steady and time dependent flow at low reynolds numbers

immersed boundary smooth extension the immersed boundary method is a simple efficient and robust numerical scheme for solving pde in general domains yet it only achieves first order spatial accuracy near embedded boundaries in this paper we introduce a new high order numerical method which we call the immersed boundary smooth extension ibse method the ibse method achieves high order accuracy by smoothly extending the unknown solution of the pde from a given smooth domain to a larger computational domain enabling the use of simple cartesian grid discretizations e g fourier spectral methods the method preserves much of the flexibility and robustness of the original ib method in particular it requires minimal geometric information to describe the boundary and relies only on convolution with regularized delta functions to communicate information between the computational grid and the boundary we present a fast algorithm for solving elliptic equations which forms the basis for simple high order implicit time methods for parabolic pde and implicit explicit methods for related nonlinear pde we apply the ibse method to solve the poisson heat burgers and fitzhugh nagumo equations and demonstrate fourth order pointwise convergence for dirichlet problems and third order pointwise convergence for neumann problems

an implicit boundary approach for viscous compressible high reynolds flows using a hybrid remeshed particle hydrodynamics method abstract we present an implicit boundary particle method with background mesh adaptation we use a brinkman penalisation to represent the boundary of the domain and a remeshed particle method to simulate viscous flow with high reynolds numbers a penalty term is added to the navier stokes equations to impose the boundary conditions the boundary conditions are enforced to a specific precision with no need to modify the numerical method or change the grid achieving an implicit approach for flow around complex boundaries geometries simulation the main idea of the brinkman penalisation method is to model the solid obstacle as a porous medium the governing equations for the compressible fluid and penalised navier stokes for the porous medium are solved simultaneously without need for interface conditions the accuracy of the method is tested for a number of benchmark problems starting with simple cases such as a periodic laminar flow in a channel poiseuille flow to more complex problems such as the lid driven cavity with high reynolds numbers the reflection and propagation of a compressible shock wave and finally swirling flow in a model two stroke diesel engine the remeshed particle mesh method coupled with brinkman penalisation provides a good quality simulation and the results are in agreement with analytical or reference solutions

a characteristic based volume penalization method for general evolution problems applied to compressible viscous flows in order to introduce solid obstacles into flows several different methods are used including volume penalization methods which prescribe appropriate boundary conditions by applying local forcing to the constitutive equations one well known method is brinkman penalization which models solid obstacles as porous media while it has been adapted for compressible incompressible viscous and inviscid flows it is limited in the types of boundary conditions that it imposes as are most volume penalization methods typically approaches are limited to dirichlet boundary conditions in this paper brinkman penalization is extended for generalized neumann and robin boundary conditions by introducing hyperbolic penalization terms with characteristics pointing inward on solid obstacles this characteristic based volume penalization cbvp method is a comprehensive approach to conditions on immersed boundaries providing for homogeneous and inhomogeneous dirichlet neumann and robin boundary conditions on hyperbolic and parabolic equations this cbvp method can be used to impose boundary conditions for both integrated and non integrated variables in a systematic manner that parallels the prescription of exact boundary conditions furthermore the method does not depend upon a physical model as with porous media approach for brinkman penalization and is therefore flexible for various physical regimes and general evolutionary equations here the method is applied to scalar diffusion and to direct numerical simulation of compressible viscous flows with the navier stokes equations both homogeneous and inhomogeneous neumann boundary conditions are demonstrated through external flow around an adiabatic and heated cylinder theoretical and numerical examination shows that the error from penalized neumann and robin boundary conditions can be rigorously controlled through an a priori penalization parameter h the error on a transient boundary is found to converge as o h which is more favorable than the error convergence of the already established dirichlet boundary condition

log depth circuits for division and related problems we present optimal depth boolean circuits depth o log n for integer division powering and multiple products we also show that these three problems are of equivalent uniform depth and space complexity in addition we describe an algorithm for testing divisibility that is optimal for both depth and space

topology optimization for unsteady flow a computational methodology for optimizing the conceptual layout of unsteady flow problems at low reynolds numbers is presented the geometry of the design is described by the spatial distribution of a fictitious material with continuously varying porosity the flow is predicted by a stabilized finite element formulation of the incompressible navier stokes equations a brinkman penalization is used to enforce zero velocities in solid material the resulting parameter optimization problem is solved by a non linear programming method the paper studies the feasibility of the material interpolation approach for optimizing the topology of unsteady flow problems the derivation of the governing equations and the adjoint sensitivity analysis are presented a design dependent stabilization scheme is introduced to mitigate numerical instabilities in porous material the emergence of non physical artifacts in the optimized material distribution is observed and linked to an insufficient resolution of the flow field and an improper representation of the pressure field within solid material by the brinkman penalization two numerical examples demonstrate that the designs optimized for unsteady flow differ significantly from their steady state counterparts copyright 2011 john wiley sons ltd

algorithms and hardness results for parallel large margin learning we consider the problem of learning an unknown large margin halfspace in the context of parallel computation giving both positive and negative results as our main positive result we give a parallel algorithm for learning a large margin half space based on an algorithm of nesterov s that performs gradient descent with a momentum term we show that this algorithm can learn an unknown margin halfspace over n dimensions using n poly 1 processors and running in time o 1 o log n in contrast naive parallel algorithms that learn a margin halfspace in time that depends polylogarithmically on n have an inverse quadratic running time dependence on the margin parameter our negative result deals with boosting which is a standard approach to learning large margin halfspaces we prove that in the original pac framework in which a weak learning algorithm is provided as an oracle that is called by the booster boosting cannot be parallelized more precisely we show that if the algorithm is allowed to call the weak learner multiple times in parallel within a single boosting stage this ability does not reduce the overall number of successive stages of boosting needed for learning by even a single stage our proof is information theoretic and does not rely on unproven assumptions

a fictitious domain approach with spread interface for elliptic problems with general boundary conditions the aim of this article is to solve second order elliptic problems in an original physical domain using a fictitious domain method with a spread interface approach the main idea of the fictitious domain approach consists in immersing the original domain of study into a geometrically bigger and simpler one called fictitious domain as the spatial discretization is being performed in the fictitious domain this method allows the use of structured meshes the discretization is not boundary fitted to the original physical domain this paper describes several ways to impose dirichlet robin or neumann boundary conditions on a spread immersed interface without locally modifying the numerical scheme and without using lagrange multipliers the numerical applications focus on difiusion and convection problems in the unit disk with dirichlet or robin boundary conditions for such problems analytical solutions can be determined for a correctly chosen source term the numerical resolution is performed using a q1 finite element scheme the spread interface approach is then combined with a local adaptive mesh refinement algorithm in order to increase the precision in the vinicity of the immersed boundary the l2 norm of the errors is computed in order to evaluate the capability of the method immersed boundaries are found in many industrial applications like two phase flow simulations fluid structure interaction etc this article represents a first step towards the simulation of these kinds of 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

convergence analysis of the q1 finite element method for elliptic problems with non boundary fitted meshes summary the aim of this paper is to derive a priori error estimates when the mesh does not fit the original domain s boundary this problematic of the last century e g the finite difference methodology returns to topical studies with the huge development of domain embedding fictitious domain or cartesiangrid methods these methods use regular structured meshes most often cartesian for non aligned domains although non boundary fitted approaches become more and more applied very few studies are devoted to theoretical error estimates in this paper the convergence of a q1 nonconforming finite element method is analyzed for secondorder elliptic problems with dirichlet robin or neumann boundary conditions the finite element method uses standard q1 rectangular finite elements as the finite element approximate space is not contained in the original solution space this method is referred to as nonconforming a stairstep boundary defined from the cartesian mesh approximates the original domain s boundary the convergence analysis of the finite element method for such a kind of non boundary fitted stairstepped approximation is no treated in the literature the study of dirichlet problems is based on similar techniques as those classically used with boundary fitted linear triangular finite elements the estimates obtained for robin problems are novel and use some more technical arguments the rate of convergence is proved to be in o h 1 2 for the h 1 norm for all general boundary conditions and classical duality arguments allow to obtain an o h error estimate in the l 2 norm for dirichlet problems numerical results obtained with fictitious domain techniques that impose original boundary conditions on a non boundary fitted approximate immersed interface are presented these results confirm the theoretical rates of convergence copyright c 2007 john wiley sons ltd

final states of decaying 2d turbulence in bounded domains influence of the geometry abstract direct numerical simulations of two dimensional decaying turbulence in wall bounded domains are presented the navier stokes equations are solved using a fourier pseudo spectral method with volume penalization starting from random initial conditions we study the influence of the geometry of the domain on the flow dynamics in particular on the long time behaviour circular square triangular and annular domains are considered and we show how the geometry plays a crucial role regarding the decay scenario towards final states three stages can be distinguished formation of coherent vortices from random initial conditions vortex wall interactions and finally relaxation towards a quasi steady structure the eigenvalues estimated from the decay rate of both energy and enstrophy depend on the geometry and agree well with the theoretical eigenvalues based on the stokes mode of the corresponding domain for the final states we find a linear functional relation between vorticity and streamfunction

a discontinuous galerkin based immersed boundary method a numerical method to approximate partial differential equations on meshes that do not conform to the domain boundaries is introduced the proposed method is conceptually simple and free of user defined parameters starting with a conforming finite element mesh the key ingredient is to switch those elements intersected by the dirichlet boundary to a discontinuous galerkin approximation and impose the dirichlet boundary conditions strongly by virtue of relaxing the continuity constraint at those elements boundary locking is avoided and optimal order convergence is achieved this is shown through numerical experiments in reaction diffusion problems copyright 2008 john wiley sons ltd

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