accurate projection methods for the incompressible navier stokes equations abstract this paper considers the accuracy of projection method approximations to the initial boundary value problem for the incompressible navier stokes equations the issue of how to correctly specify numerical boundary conditions for these methods has been outstanding since the birth of the second order methodology a decade and a half ago it has been observed that while the velocity can be reliably computed to second order accuracy in time and space the pressure is typically only first order accurate in the l norm this paper identifies the source of this problem in the interplay of the global pressure update formula with the numerical boundary conditions and presents an improved projection algorithm which is fully second order accurate as demonstrated by a normal mode analysis and numerical experiments in addition a numerical method based on a gauge variable formulation of the incompressible navier stokes equations which provides another option for obtaining fully second order convergence in both velocity and pressure is discussed the connection between the boundary conditions for projection methods and the gauge method is explained in detail

a finite difference galerkin formulation for the incompressible navier stokes equations abstract the development of a new computational method for solving the incompressible navier stokes equations in primitive variable form is presented it is found that certain finite difference approximations for these equations can be transformed into an equivalent system which efficiently determines the discrete velocity field and which completely eliminates the pressure two such difference schemes for two dimensional problems are examined and some preliminary numerical results are discussed for the steady driven cavity problem

an unsplit higher order godunov method for scalar conservation laws in multiple dimensions abstract in this paper we develop an unsplit higher order godunov method for scalar conservation laws in two dimensions the method represents an extension for the special case being considered of methods developed by colella and by van leer in our method we begin with a piecewise bilinear representation of the solution on each grid cell a new piecewise bilinear representation at the next time level is then obtained from a two step procedure in the first step a conservative predictor corrector scheme derived from the integral form of the differential equation and characteristic considerations is used to obtain average values of the solution over grid cells at the new time level next these new average values are then used to construct a limited piecewise bilinear profile for each cell at the new time level the resulting method is shown to satisfy a maximum principle for constant coefficient linear advection computational results are presented comparing the new method to colella s method for linear advection the method is also applied to two model problems from porous media flow miscible and immiscible displacement the new scheme provides accurate resolution of sharp fronts without any significant distortion

effects of the computational time step on numerical solutions of turbulent flow effects of large computational time steps on the computed turbulence were investigated using a fully implicit method in turbulent channel flow computations the largest computational time step in wall units which led to accurate prediction of turbulence statistics was determined turbulence fluctuations could not be sustained if the computational time step was near or larger than the kolmogorov time scale

stable and accurate pressure approximation for unsteady incompressible viscous flow how to properly specify boundary conditions for pressure is a longstanding problem for the incompressible navier stokes equations with no slip boundary conditions an analytical resolution of this issue stems from a recently developed formula for the pressure in terms of the commutator of the laplacian and leray projection operators here we make use of this formula to a improve the accuracy of computing pressure in two kinds of existing time discrete projection methods implicit in viscosity only and b devise new higher order accurate time discrete projection methods that extend a slip correction idea behind the well known finite difference scheme of kim and moin we test these schemes for stability and accuracy using various combinations of c 0 finite elements for all three kinds of time discretization one can obtain third order accuracy for both pressure and velocity without a time step stability restriction of diffusive type furthermore two kinds of projection methods are found stable using piecewise linear elements for both velocity and pressure

application of a fractional step method to incompressible navier stokes equations a numerical method for computing three dimensional time dependent incompressible flows is presented the method is based on a fractional step or time splitting scheme in conjunction with the approximate factorization technique it is shown that the use of velocity boundary conditions for the intermediate velocity field can lead to inconsistent numerical solutions appropriate boundary conditions for the intermediate velocity field are derived and tested numerical solutions for flows inside a driven cavity and over a backward facing step are presented and compared with experimental data and other numerical results

the accuracy of the fractional step method we analyze the accuracy of the fractional step method of kim and moin j comput phys 59 1985 pp 308 323 for the incompressible navier stokes equations we show that the boundary conditions cannot be exactly satisfied in the projection step and that this limits the accuracy of the method we also show that the pressure in any projection method can be at best first order accurate our analysis is simpler and more direct than the previous analyses of this method we also show that there is no numerical boundary layer for velocity or pressure but there is one for the auxiliary pressure variable

a quadratic spline based interface quasi reconstruction algorithm for accurate tracking of two phase flows a new quadratic spline based interface quasi reconstruction algorithm is presented which provides an accurate and continuous representation of the interface in a multiphase domain and facilitates the direct estimation of local interfacial curvature the fluid interface in each of the mixed cells is represented by piecewise parabolic curves and an initial discontinuous plic approximation of the interface is progressively converted into a smooth quadratic spline made of these parabolic curves the conversion is achieved by a sequence of predictor corrector operations enforcing function c 0 and derivative c 1 continuity at the cell boundaries using simple analytical expressions for the continuity requirements the efficacy and accuracy of the current algorithm has been demonstrated using standard test cases involving reconstruction of known static interface shapes and dynamically evolving interfaces in prescribed flow situations these benchmark studies illustrate that the present algorithm performs excellently as compared to the other interface reconstruction methods available in literature quadratic rate of error reduction with respect to grid size has been observed in all the cases with curved interface shapes only in situations where the interface geometry is primarily flat the rate of convergence becomes linear with the mesh size the flow algorithm implemented in the current work is designed to accurately balance the pressure gradients with the surface tension force at any location as a consequence it is able to minimize spurious flow currents arising from imperfect normal stress balance at the interface this has been demonstrated through the standard test problem of an inviscid droplet placed in a quiescent medium finally the direct curvature estimation ability of the current algorithm is illustrated through the coupled multiphase flow problem of a deformable air bubble rising through a column of water

on convergence of block centered finite differences for elliptic problems we consider linear selfadjoint elliptic problems with neumann boundary conditions in rectangular domains we demonstrate that with sufficiently smooth data the discrete l 2 norm errors for tensor product block centered finite differences in both the approximate solution and its first derivatives are second order for all nonuniform grids extensions to nonselfadjoint and parabolic problems are discussed

application of lattice boltzmann method to sensitivity analysis via complex differentiation sensitivity analysis of flows computed by lattice boltzmann method via complex differentiation is proposed the theoretical work is illustrated and the proposed method assessed considering the d2q9 scheme along with the differentiation of the lattice boltzmann equation boundary condition implementation is also detailed some examples illustrate the capability of the proposed method

a cartesian grid embedded boundary method for hyperbolic conservation laws we present a second order godunov algorithm to solve time dependent hyperbolic systems of conservation laws on irregular domains our approach is based on a formally consistent discretization of the conservation laws on a finite volume grid obtained from intersecting the domain with a cartesian grid we address the small cell stability problem associated with such methods by hybridizing our conservative discretization with a stable nonconservative discretization at irregular control volumes and redistributing the difference in the mass increments to nearby cells in a way that preserves stability and local conservation the resulting method is second order accurate in l 1 for smooth problems and is robust in the presence of large amplitude discontinuities intersecting the irregular boundary

a gradient augmented level set method for unstructured grids we present and test a gradient augmented level set method for unstructured grids which has been designed to allow for easy integration in parallel flow solvers the means to this end is a hermite formulation where the gradient is stored and advected as an independent variable along with the level set function the method uses narrow band storage in conjunction with a fast marching method which is also designed to take advantage of the gradient augmentation we demonstrate through standard passive advection test cases that the proposed method is able to compute interface motion with performance comparable to popular high performing methods on structured grids

on error estimates of the projection methods for the navier stokes equations second order schemes we present in this paper a rigorous error analysis of several projection schemes for the approximation of the unsteady incompressible navier stokes equations the error analysis is accomplished by interpreting the respective projection schemes as second order time discretizations of a perturbed system which approximates the navier stokes equations numerical results in agreement with the error analysis are also presented

a new family of 5 5 cc 4oc schemes applicable for unsteady navier stokes equations in this paper a new family of implicit compact finite difference schemes for computation of unsteady convection diffusion equation with variable convection coefficient is proposed the schemes which are fourth order accurate in space and second or lower order accurate in time depending on the choice of weighted time average parameter are then applied to unsteady navier stokes system the proposed schemes where transport variable and its first derivatives are carried as the unknowns combine virtues of compact discretization and pade scheme for spatial derivative these schemes which are based on a five point stencil with constant coefficients named as 5 5 constant coefficient 4th order compact 5 5 cc 4oc give rise to a diagonally dominant system of equations and shows higher accuracy and better phase and amplitude error characteristics than some of the standard methods these schemes are capable of using a grid aspect ratio other than unity and are unconditionally stable they efficiently capture both transient and steady solutions of linear and nonlinear convection diffusion equations with dirichlet as well as neumann boundary conditions subsequently the proposed schemes are applied to problems governed by the incompressible navier stokes equations the results obtained are in excellent agreement with analytical and available numerical results in all cases establishing efficiency and accuracy of the proposed schemes

pressure stability in fractional step finite element methods for incompressible flows abstract the objective of this paper is to analyze the pressure stability of fractional step finite element methods for incompressible flows that use a pressure poisson equation for the classical first order projection method it is shown that there is a pressure control which depends on the time step size and therefore there is a lower bound for this time step for stability reasons the situation is much worse for a second order scheme in which part of the pressure gradient is kept in the momentum equation the pressure stability in this case is extremely weak to overcome these shortcomings a stabilized fractional step finite element method is also considered and its stability is analyzed some simple numerical examples are presented to support the theoretical results

review and unification of methods for computing derivatives of multidisciplinary computational models this paper presents a review of all existing discrete methods for computing the derivatives of computational models within a unified mathematical framework this framework hinges on a new equation the unifying chain rule from which all the methods can be derived the computation of derivatives is described as a two step process the evaluation of the partial derivatives and the computation of the total derivatives which are dependent on the partial derivatives finite differences the complex step method and symbolic differentiation are discussed as options for computing the partial derivatives it is shown that these are building blocks with which the total derivatives can be assembled using algorithmic differentiation the direct and adjoint methods and coupled analytic methods for multidisciplinary systems each of these methods is presented and applied to a common numerical example to demonstrate and compare the various approaches the paper ends with a discussion of current challenges and possib

direct simulations of turbulent flow using finite difference schemes abstract this paper presents finite difference solutions to the evolution of small amplitude disturbances and incompressible fully developed turbulent channel flow the main objective of the paper is to provide a comprehensive comparison between the results obtained using finite difference and spectral methods an advantage of finite difference schemes over the highly accurate spectral methods lies in the ease with which they can be applied to complex geometries the finite difference methods used include a kinetic energy conserving type of central difference scheme and a high order accurate upwind difference scheme unlike the central difference scheme the upwind difference scheme was found not to require a kinetic energy conservation property to control aliasing error the dissipative nature of the upwind scheme results in a damping of the higher frequency content as a result very little energy is aliased back the computed data including first and second order statistics for the turbulent channel flow case are found to compare well with experimental data and earlier spectral simulations it appears that the high order accurate upwind scheme is a good candidate for direct simulations of turbulent flows over complex geometries

an accurate second order approximation factorization method for time dependent incompressible navier stokes equations in spherical polar coordinates a finite difference method for solving three dimensional time dependent incompressible navier stokes equations in spherical polar coordinates is presented in detail a new algorithm which is second order accurate in time and space is considered and decoupling between the velocity and the pressure is achieved by this algorithm further the numerical method is tested by computing the spherical couette flow between two concentric spheres with the inner one rotating a comparison of the numerical solutions with available numerical results and experimental measurements is made it is demonstrated that the numerical code is valid for solving three dimensional unsteady incompressible navier stokes equations in spherical polar coordinates

an accurate adaptive solver for surface tension driven interfacial flows a method combining an adaptive quad octree spatial discretisation geometrical volume of fluid interface representation balanced force continuum surface force surface tension formulation and height function curvature estimation is presented the extension of these methods to the quad octree discretisation allows adaptive variable resolution along the interface and is described in detail the method is shown to recover exact equilibrium to machine accuracy between surface tension and pressure gradient in the case of a stationary droplet irrespective of viscosity and spatial resolution accurate solutions are obtained for the classical test case of capillary wave oscillations an application to the capillary breakup of a jet of water in air further illustrates the accuracy and efficiency of the method the source code of the implementation is freely available as part of the gerris flow solver

aerodynamic optimization algorithm with integrated geometry parameterization and mesh movement an efficient gradient based algorithm for aerodynamic shape optimization is presented the algorithm consists of several components including a novel integrated geometry parameterization and mesh movement a parallel newton krylov flow solver and an adjoint based gradient evaluation to integrate geometry parameterization and mesh movement generalized b spline volumes are used to parameterize both the surface and volume mesh the volume mesh of b spline control points mimics a coarse mesh a linear elasticity mesh movement algorithm is applied directly to this coarse mesh and the fine mesh is regenerated algebraically using this approach mesh movement time is reduced by two to three orders of magnitude relative to a node based movement the mesh adjoint system also becomes smaller and is thus amenable to complex step derivative approximations when solving the flow adjoint equations using restarted krylov subspace methods a nested subspace strategy is shown to be more robust than truncating the entire subspace optimization is accomplished using a sequential quadratic programming algorithm the effectiveness of the complete algorithm is demonstrated using a lift constrained induced drag minimization that involves large changes in geometry

efficient implementation of thinc scheme in this paper we improve the multi dimensional thinc tangent of hyperbola for interface capturing scheme f xiao y honma t kono a simple algebraic interface capturing scheme using hyperbolic tangent function int j numer meth fluid 48 2005 1023 the thinc scheme is a vof volume of fluid type method in the original thinc scheme one dimensional thinc scheme was straightforwardly used for multi dimensional cases in this paper we propose the wlic weighed line interface calculation method to extend the thinc scheme to multi dimension in the wlic method the interface is reconstructed by taking an average of line interfaces along x y and z coordinates with the weights calculated from surface normal the wlic method can extend the thinc scheme to multi dimension while maintaining simplicity of implementation and achieve a higher accuracy than the original thinc scheme the wlic method can readily extend the thinc scheme to three dimension

multicomplex newmark beta time integration method for sensitivity analysis in structural dynamics sensitivity analysis of structural systems is important for identifying important parameters that influence the dynamic response of a model this paper describes an extension of the standard newmark beta algorithm to the multicomplex mathematical domain such that time dependent high order high accuracy derivatives of dynamic systems can be obtained along with the traditional response in particular complex and bicomplex analyses were used to compute accurate first and second order derivatives of several example problems these derivatives were shown to be highly accurate when compared to analytical and finite difference estimates the method is general and any order derivative including mixed derivatives can be computed

discrete shape sensitivity equations for aerodynamic problems intensive research and development in computational fluid dynamics cfd has recently produced many powerful cfd codes to simulate complex aerodynamic phenomena however in order to enhance the usefulness of these cfd codes for design practice development of design sensitivity equations compatible to these codes becomes very important this paper represents a part of such an effort to develop a sensitivity analysts methodology that enables the sensitivity equations to be implemented into existing cfd codes with minimal code modification the methodology is based upon a pre elimination procedure which accounts for consistently linearized boundary conditions formulations of both the direct differentiation and the adjoint variable methods will be presented in the paper

implicit multigrid algorithms for incompressible turbulent flows on unstructured grids an implicit code for computing inviscid and viscous incompressible flows on unstructured grids is described the foundation of the code is a backward euler time discretization for which the linear system is approximately solved at each time step with either a point implicit method or a preconditioned generalized minimal residual gmres technique for the gmres calculations several techniques are investigated for forming the matrix vector product convergence acceleration is achieved through a multigrid scheme that uses nonnested coarse grids that are generated using a technique described in the present paper convergence characteristics are investigated and results are compared with an exact solution for the inviscid flow over a four element airfoil viscous results which are compared with experimental data include the turbulent flow over a naca 4412 airfoil a three element airfoil for which mach number effects are investigated and three dimensional flow over a wing with a partial span flap

approximate factorization as a high order splitting for the implicit incompressible flow equations abstract we apply the method of approximate factorization to derive a second order accurate splitting of the incompressible flow equations stokes or navier stokes equations this is novel because the method of approximate factorization was believed inapplicable to this type of equation system we demonstrate the resulting splitting on a second order crank nicolson discretization and point out its intimate relationship to some existing second order accurate splitting projection methods further we use the approximate factorization method to derive entirely new splittings we first develop a new generalized second order accurate splitting which may be specialized to a variety of applications we indicate its application to finite elements checker board free cell centered discretizations and ocean modeling we then generalized the original splitting to an arbitrarily high order scheme

a high order discontinuous galerkin method for 2d incompressible flows in this paper we introduce a high order discontinuous galerkin method for two dimensional incompressible flow in the vorticity stream function formulation the momentum equation is treated explicitly utilizing the efficiency of the discontinuous galerkin method the stream function is obtained by a standard poisson solver using continuous finite elements there is a natural matching between these two finite element spaces since the normal component of the velocity field is continuous across element boundaries this allows for a correct upwinding gluing in the discontinuous galerkin framework while still maintaining total energy conservation with no numerical dissipation and total enstrophy stability the method is efficient for inviscid or high reynolds number flows optimal error estimates are proved and verified by numerical experiments

fully conservative higher order finite difference schemes for incompressible flow conservation properties of the mass momentum and kinetic energy equations for incompressible flow are specified as analytical requirements for a proper set of discrete equations existing finite difference schemes in regular and staggered grid systems are checked for violations of the conservation requirements and a few important discrepancies are pointed out in particular it is found that none of the existing higher order schemes for a staggered mesh system simultaneously conserve mass momentum and kinetic energy this deficiency is corrected through the derivation of a general family of fully conservative higher order accurate finite difference schemes for staggered grid systems finite difference schemes in a collocated grid system are also analyzed and a violation of kinetic energy conservation is revealed the predicted conservation properties are demonstrated numerically in simulations of inviscid white noise performed in a two dimensional periodic domain the proposed fourth order schemes in a staggered grid system are generalized for the case of a non uniform mesh and the resulting scheme is used to perform large eddy simulations of turbulent channel flow

the immersed boundary method for advection electrodiffusion with implicit timestepping and local mesh refinement we describe an immersed boundary method for problems of fluid solute structure interaction the numerical scheme employs linearly implicit timestepping allowing for the stable use of timesteps that are substantially larger than those permitted by an explicit method and local mesh refinement making it feasible to resolve the steep gradients associated with the space charge layers as well as the chemical potential which is used in our formulation to control the permeability of the membrane to the possibly charged solute low reynolds number fluid dynamics are described by the time dependent incompressible stokes equations which are solved by a cell centered approximate projection method the dynamics of the chemical species are governed by the advection electrodiffusion equations and our semi implicit treatment of these equations results in a linear system which we solve by gmres preconditioned via a fast adaptive composite grid fac solver numerical examples demonstrate the capabilities of this methodology as well as its convergence properties

a stable and convergent scheme for viscoelastic flow in contraction channels we present a new algorithm to simulate unsteady viscoelastic flows in abrupt contraction channels in our approach we split the viscoelastic terms of the oldroyd b constitutive equation using duhamel s formula and discretize the resulting pdes using a semi implicit finite difference method based on a lax wendroff method for hyperbolic terms in particular we leave a small residual elastic term in the viscous limit by design to make the hyperbolic piece well posed a projection method is used to impose the incompressibility constraint we are able to compute the full range of unsteady elastic flows in an abrupt contraction channel from the viscous limit to the elastic limit in a stable and convergent manner we demonstrate the range of our method for unsteady flow of a maxwell fluid with and without viscosity in planar contraction channels we also demonstrate stable and convergent results for benchmark high weissenberg number problems at we 1 and we 10

verified predictions of shape sensitivities in wall bounded turbulent flows by an adaptive finite element method a continuous sensitivity equation cse method is presented for shape parameters in turbulent wall bounded flows modeled with the standard email protected turbulence model with wall functions differentiation of boundary conditions and their complex dependencies on shape parameters including the two velocity scale wall functions is presented in details along with the appropriate methodology required for the cse method to ensure accuracy grid convergence and to reduce computational time an adaptive finite element method driven by asymptotically exact error estimations is used the adaptive process is controlled by error estimates on both flow and sensitivity solutions firstly the proposed approach is applied on a problem with a closed form solution derived using the method of the manufactured solution to perform code verification results from adaptive grid refinement studies show verification of flow and sensitivity solvers error estimators and the adaptive strategy secondly we consider turbulent flows around a square cross section cylinder in proximity of a solid wall we examine the quality of the numerical solutions by performing solution verification and validation then sensitivity analysis of these turbulent flows is performed to investigate the ability of the method to deal with non trivial geometrical changes sensitivity information is used to estimate uncertainties in the flow solution caused by uncertainties in the shape parameter and to perform fast evaluation of flows on nearby configurations

discrete exterior calculus discretization of incompressible navier stokes equations over surface simplicial meshes a conservative discretization of incompressible navier stokes equations is developed based on discrete exterior calculus dec a distinguishing feature of our method is the use of an algebraic discretization of the interior product operator and a combinatorial discretization of the wedge product the governing equations are first rewritten using the exterior calculus notation replacing vector calculus differential operators by the exterior derivative hodge star and wedge product operators the discretization is then carried out by substituting with the corresponding discrete operators based on the dec framework numerical experiments for flows over surfaces reveal a second order accuracy for the developed scheme when using structured triangular meshes and first order accuracy for otherwise unstructured meshes by construction the method is conservative in that both mass and vorticity are conserved up to machine precision the relative error in kinetic energy for inviscid flow test cases converges in a second order fashion with both the mesh size and the time step

optimal shape design of a two dimensional asymmetric diffuser in turbulent flow an optimal shape of two dimensional asymmetric diffuser with maximum pressure recovery at the exit is numerically obtained using a mathematical theory based on the variational calculus and gradient algorithm the initial diffuser is taken to be a two dimensional asymmetric diffuser for which much experimental and numerical data are available the reynolds number based on the bulk mean velocity and the channel height at the diffuser entrance is 1 8 10 4 f rom this initial shape optimal diffuser shapes are designed for six different geometric constraints such as the streamwise length and height of the diffuser the optimality condition for maximum pressure recovery is obtained to be zero skin friction along the diffuser wall the turbulent flow inside the diffuser is predicted using the k v2 f model and optimal shapes are obtained through iterative procedures to satisfy the optimality condition with the shape design flow separation that appeared in the initial diffuser is completely removed or significantly reduced for one of the optimal diffuser shapes obtained large eddy simulation is carried out to validate the result of the shape design the wall shear stress wall pressure mean velocity and turbulence quantities obtained from large eddy simulation are in good agreement with those from the simulation using the turbulence model

efficient algorithm on a nonstaggered mesh for simulating rayleigh benard convection in a box an efficient semi implicit second order accurate finite difference method is described for studying incompressible rayleigh benard convection in a box with sidewalls that are periodic thermally insulated or thermally conducting operator splitting and a projection method reduce the algorithm at each time step to the solution of four helmholtz equations and one poisson equation and these are are solved by fast direct methods the method is numeri cally stable even though all field values are placed on a single non staggered mesh commensurate with the boundaries the efficiency and accuracy of the method are characterized for several representative convection problems

causality and model abstraction abstract much of science and engineering is concerned with characterizing processes by equations describing the relations that hold among parameters of objects and govern their behavior over time in formal descriptions of processes in terms of parameters and equations the notion of causality is rarely made explicit formal treatments of the foundations of sciences have avoided discussions of causation and spoken only of functional relations among variables nevertheless the notion of causality plays an important role in our understanding of phenomena even when we describe the behavior of a system formally in terms of acausal mathematical relations we often give an informal intuitive explanation of why the system behaves the way it does in terms of cause effect relations in this paper we will present an operational definition of causal ordering the definition allows us to extract causal dependency relations among variables implicit in a model of a system when a model is represented as a set of acausal mathematical relations our approach is based on the theory of causal ordering first presented by simon 22 the paper shows how to use the theory and its extension in reasoning about physical systems further the paper studies the relation of the theory to the problems of model aggregation

a second order projection method for the incompressible navier stokes equations in this paper we describe a second order projection method for the time dependent incom pressible navier stokes equations as in the original projection method developed by chorin we first solve diffusion convection equations to predict intermediate velocities which are then projected onto the space of divergence free vector fields by introducing more coupling between the diffusion onvection step and the projection step we obtain a temporal discretiza tion that is second order accurate our treatment of the diffusion convection step uses a specialized higher order godunov method for differencing the nonlinear convective terms that provides a robust treatment of these terms at high reynolds number the godunov procedure is second order accurate for smooth flow and remains stable for discontinuous initial data even in the zero viscosity limit we approximate the projection directly using a galerkin procedure that uses a local basis for discretely divergence free vector fields numerical results are presented validating the convergence properties of the method we also apply the method to doubly periodic shear layers to assess the performance of the method on more difficult

physically motivated domain decomposition for singularly perturbed equations a domain decomposition algorithm suitable for the efficient and accurate solution of a parabolic reaction convection diffusion equation with small parameter multiplying the diffusion term is presented convergence is established via maximum principle arguments the equation arises in the modeling of laminar transonic flow decomposition into subdomains is accomplished via singular perturbation analysis which dictates regions where certain reduced equations may be solved in place of the full equation effectively preconditioning the problem this paper concentrates on the theoretical basis of the method establishing local and global a priori error bounds

quaternion and octonion based finite element analysis methods for computing multiple first order derivatives abstract the complex taylor series expansion method for computing accurate first order derivatives is extended in this work to quaternion octonion and any order cayley dickson algebra the advantage of this new approach is that highly accurate multiple first order derivatives can be obtained in a single analysis quaternion and octonion based finite element analysis methods were developed in order to compute up to three quaternion and up to seven octonion first order derivatives of shape material properties and or loading conditions in a single analysis the traditional finite element formulation was modified such that each degree of freedom was augmented with three or seven additional imaginary nodes the quaternion and octonion based methods were integrated within the abaqus commercial finite element code through a user element subroutine numerical examples are presented for thermal conductivity and linear elasticity however the methodology is general the results indicate that the quaternion and octonion based methods provide derivatives of the same high accuracy as the complex finite element method but are significantly more efficient a fortran code to solve a simple seven variable quaternion example is given in the appendix

algorithm developments for discrete adjoint methods this paper presents a number of algorithm developments for adjoint methods using the discrete approach in which the discretisation of the non linear equations is linearised and the resulting matrix is then transposed with a new iterative procedure for solving the adjoint equations exact numerical equivalence is maintained between the linear and adjoint discretisations the incorporation of strong boundary conditions within the discrete approach is discussed as well as a new application of adjoint methods to linear unsteady flow in turbomachinery

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

geometry parameterization method for multidisciplinary applications a technique is described for parameterizing geometries in a manner suitable for numerical simulations used in design optimization the new methodology referred to as control grids is flexible easy to use extendable and applicable to multidisciplinary design optimization although significantly different than free form deformation the present method provides many similar capabilities comparisons between the current technique and free form deformation are provided to highlight the advantages and disadvantages of each method the new parameterization technology is described and demonstrated in both two and three dimensions for single and multidisciplinary applications

an adjoint based lattice boltzmann method for noise control problems in this paper optimal control of acoustic problems is addressed within the lattice boltzmann method framework to this end an adjoint based lattice boltzmann method is proposed to solve the adjoint problem the adjoint state provides an easy access to the optimization gradients the line search step in newton s descent method is performed through a combination of complex differentiation and adjoint problem in the lattice boltzmann method the implementation of an active noise reduction method for two dimensional weakly compressible low mach number flows is discussed and the applicability of the method is assessed

sensitivity derivatives for advanced cfd algorithm and viscous modeling parameters via automatic differentiation the computational technique of automatic differentiation ad is applied to a complicated computer program to illustrate the simplicity efficiency and versatility of ad with complex algorithms for use within a sensitivity analysis many algorithmic and physics modeling coefficients appear in computer programs that are routinely set in an ad hoc manner ad can be used to enhance computer programs with derivative information suitable for guiding formal sensitivity analyses which allows these coefficient values to be chosen in a rigorous manner to achieve particular program properties such as an improved convergence rate or improved accuracy in this paper ad is applied to a three dimensional thin layer navier stokes multigrid flow solver to assess the feasibility and computational impact of obtaining exact sensitivity derivatives with respect to algorithmic and physics modeling parameters typical of those needed for sensitivity analyses calculations are performed for an onera m6 wing in transonic flow with both the baldwin lomax and johnson king turbulence models the wing lift drag and pitching moment coefficients are differentiated with respect to two different groups of input parameters the first group consists of the second and fourth order damping coefficients of the computational algorithm whereas the second group consists of two parameters in the viscous turbulent flow physics modeling results obtained via ad are compared for both accuracy and computational efficiency with the results obtained with divided differences dd the ad results are accurate extremely simple to obtain and show significant computational advantage over those obtained by dd for some cases

using an adjoint approach to eliminate mesh sensitivities in computational design an adjoint algorithm for efficiently incorporating the effects of mesh sensitivities in a computational design framework is introduced the method eliminates the need for explicit linearizations of the mesh movement scheme with respect to the geometric parameterization variables an expense that has hindered large scale design optimization for practical applications the effects of the mesh sensitivities can be accounted for through the solution of an adjoint problem equivalent in cost to a single mesh movement computation followed by an explicit matrix vector product whose cost scales with the number of design variables and the resolution of the parameterized surface grid the methodology augments the current practice of using adjoints solely for the flowfield and leads to a dramatic computational savings the accuracy of the implementation is established and several sample design optimizations are shown

short note an interface capturing method with a continuous function the thinc method on unstructured triangular and tetrahedral meshes a novel interface capturing method is proposed to compute moving interfaces on unstructured grids with triangular 2d and tetrahedral 3d elements different from the conventional vof volume of fluid method which involves geometric reconstructions of the interface the present method is based on the algebraic reconstruction approach originally developed in the thinc tangent of hyperbola interface capturing scheme by xiao et al 2005 17 a continuous multidimensional hyperbolic tangent function is employed for retrieving the jump like distribution of the indicator function which avoids the explicit geometric representation of the interface and thus substantially reduces the algorithmic complexity in unstructured grids numerical diffusion and smearing are effectively eliminated and the compact thickness of the jump transition layer in the volume fraction is retained throughout the computation even for largely deformed interface the solution quality of the present scheme is comparable to the vof method with plic piecewise linear interface calculation algorithm

a geometrical area preserving volume of fluid advection method a new class of algorithms that preserve mass exactly for incompressible flows on a cartesian mesh are presented they amount to piecewise linear area preserving mappings of tessellations of the plane they are equivalent to volume of fluid vof advection methods which are decomposed into an eulerian implicit scheme in one direction followed by a lagrangian explicit step in the other one it is demonstrated that mass conservation is exact for incompressible flows and that there are no undershoots or overshoots of the volume fraction which thus always remains constrained between 0 and 1

a cartesian grid projection method for the incompressible euler equations in complex geometries many problems in fluid dynamics require the representation of complicated internal or external boundaries of the flow here we present a method for calculating time dependent incompressible inviscid flow which combines a projection method with a cartesian grid approach for representing geometry in this approach the body is represented as an interface embedded in a regular cartesian mesh the advection step is based on a cartesian grid algorithm for compressible flow in which the discretization of the body near the flow uses a volume of fluid representation a redistribution procedure is used to eliminate time step restrictions due to small cells where the boundary intersects the mesh the projection step uses an approximate projection based on a cartesian grid method for potential flow the method incorporates knowledge of the body through volume and area fractions along with certain other integrals over the mixed cells convergence results are given for the projection itself and for the time dependent algorithm in two dimensions the method is also demonstrated on flow past a half cylinder with vortex shedding

reduced order models for parameter dependent geometries based on shape sensitivity analysis the proper orthogonal decomposition pod is widely used to derive low dimensional models of large and complex systems one of the main drawback of this method however is that it is based on reference data when they are obtained for one single set of parameter values the resulting model can reproduce the reference dynamics very accurately but generally lack of robustness away from the reference state it is therefore crucial to enlarge the validity range of these models beyond the parameter values for which they were derived this paper presents two strategies based on shape sensitivity analysis to partially address this limitation of the pod for parameters that define the geometry of the problem at hand design or shape parameters we first detail the methodology to compute both the pod modes and their lagrangian sensitivities with respect to shape parameters from them we derive improved reduced order bases to approximate a class of solutions over a range of parameter values secondly we demonstrate the efficiency and limitations of these approaches on two typical flow problems 1 the one dimensional burgers equation 2 the two dimensional flows past a square cylinder over a range of incidence angles

accurate stable and efficient navier stokes solvers based on explicit treatment of the pressure term we present numerical schemes for the incompressible navier stokes equations based on a primitive variable formulation in which the incompressibility constraint has been replaced by a pressure poisson equation the pressure is treated explicitly in time completely decoupling the computation of the momentum and kinematic equations the result is a class of extremely efficient navier stokes solvers full time accuracy is achieved for all flow variables the key to the schemes is a neumann boundary condition for the pressure poisson equation which enforces the incompressibility condition for the velocity field irrespective of explicit or implicit time discretization of the viscous term in the momentum equation the explicit time discretization of the pressure term does not affect the time step constraint indeed we prove unconditional stability of the new formulation for the stokes equation with explicit treatment of the pressure term and first or second order implicit treatment of the viscous term systematic numerical experiments for the full navier stokes equations indicate that a second order implicit time discretization of the viscous term with the pressure and convective terms treated explicitly is stable under the standard cfl condition additionally various numerical examples are presented including both implicit and explicit time discretizations using spectral and finite difference spatial discretizations demonstrating the accuracy flexibility and efficiency of this class of schemes in particular a galerkin formulation is presented requiring only c0 elements to implement

an analysis of the fractional step method the fractional step method for solving the incompressible navier stokes equations in primitive variables is analyzed as a block lu decomposition in this formulation the issues involving boundary conditions for the intermediate velocity variables and the pressure are clearly resolved in addition it is shown that poor temporal accuracy first order is not due to boundary conditions but due to the method itself a generalized block lu decomposition that overcomes this difficulty is presented allowing arbitrarily high temporal order of accuracy the generalized decomposition is shown to be useful for a wide range of problems including steady problems technical issues such as stability and the appropriate pressure update scheme are also addressed numerical simulations of the steady incompressible navier stokes equations in a square domain confirm the theoretical results 18 refs 4 figs

a cell centered adaptive projection method for the incompressible euler equations we present an algorithm to compute adaptive solutions for incompressible flows using block structured local refinement in both space and time this method uses a projection formulation based on a cell centered approximate projection which allows the use of a single set of cell centered solvers because of refinement in time additional steps are taken to accurately discretize the advection and projection operators at grid refinement boundaries using composite operators which span the coarse and refined grids this ensures that the method is approximately freestream preserving and satisfies an appropriate form of the divergence constraint

impulse response functions based on a causal approach to residual orthogonalization in vector autoregressions abstract a data determined method for testing structural models of the errors in vector autoregressions is discussed the method can easily be combined with prior economic knowledge and a subjective analysis of data characteristics to yield valuable information concerning model selection and specification in one dimension it turns out that standard t statistics can be used to test the various overidentifying restrictions that are implied by a model in another dimension the method compares a priori knowledge of a structural model for the errors with the properties exhibited by the data thus this method may help to ensure that orderings of the errors for impulse response and forecast error variance decomposition analyses are sensible given the data two economic examples are used to illustrate the method

adaptive optimization based improvement of tetrahedral meshes a tetrahedral mesh improvement method is presented that uses an optimization based node perturbation technique to minimize a cost function the cost function for each tetrahedron is based on the element condition number this condition number incorporates a weight matrix that transforms the element from physical space to a reference space the condition number is a measure of the conformity of the physical element with respect to the reference element feature based adaptation is incorporated into the element weight matrix to influence the node perturbation scheme to favor nonunit aspect ratio elements the mesh improvement scheme produces meshes that cluster grid nodes in high gradient regions of the selected adaptation function results for several inviscid three dimensional cases are included

on the generalization of the complex step method we generalize the well known complex step method for computing derivatives by introducing a complex step in a strict sense exploring different combinations of terms we derive 52 approximations for computing the first order derivatives and 43 for the second order derivatives for an appropriate combination of terms and appropriate choice of the step size in the real and imaginary directions fourth order accuracy can be achieved in a very simple and efficient scheme on a compact stencil new different ways of computing second order derivatives in one single step are shown many of the first order derivative approximations avoid the problem of subtractive cancellation inherent to the classic finite difference approximations for real valued steps and the superior accuracy and stability of the generalized complex step approximations are demonstrated for an analytic test function

using complex variables to estimate derivatives of real functions a method to approximate derivatives of real functions using complex variables which avoids the subtractive cancellation errors inherent in the classical derivative approximations is described numerical examples illustrating the power of the approximation are presented

gerris a tree based adaptive solver for the incompressible euler equations in complex geometries an adaptive mesh projection method for the time dependent incompressible euler equations is presented the domain is spatially discretised using quad octrees and a multilevel poisson solver is used to obtain the pressure complex solid boundaries are represented using a volume of fluid approach second order convergence in space and time is demonstrated on regular statically and dynamically refined grids the quad octree discretisation proves to be very flexible and allows accurate and efficient tracking of flow features the source code of the method implementation is freely available

causal ordering comparative statics and near decomposability abstract the notion of causal ordering has been explicated for systems in the form of sets of linear algebraic equations and also for dynamic models consisting of linear differential equations since we often wish to aggregate or disaggregate models of systems or to consider the static equilibrium of a dynamic model the question arises as to how these transformations from one model of a system to another affect the causal ordering of variables the present paper provides some answers to this question

on the accuracy of the complex step finite difference method abstract the complex step finite difference method csfdm is a very simple methodology that can be implemented in well known numerical techniques helping to improve for instance in the wave propagation problem time and or space derivative based wavefields we clarify differences between the csfdm and previous implementations of the complex step cs derivative approximation in well known numerical techniques we study dispersion properties for the one way and two way wave equations using the finite difference method fdm the pseudospectral method psm the finite element method fem and the csfdm under the influence of a plane wave and ricker source time functions we show the gain in numerical accuracy offered by the methodology of the csfdm over the fdm psm and fem we finally discuss consequences of the csfdm in future scenarios and propose directions of study in this area

a bridge between projection methods and simple type methods for incompressible navier stokes equations a bridge is built between projection methods and simple type methods semi implicit method for pressure linked equation a general second order accurate projection method is developed for the simulation of incompressible unsteady flows by employing a non linear update of pressure term as n pn 1 i n pn where n is a coefficient matrix which may depend on the grid size time step and even velocity it includes three and four step projection methods the standard simple method is written in a concise formula for steady and unsteady flows it is proven that simple type methods have second order temporal accuracy for unsteady flows the classical second order projection method and simple type methods are united within the framework of the general second order projection formula two iteration algorithms of simple type methods for unsteady flows are described and discussed in addition detailed formulae are provided for general projection methods by using the runge kutta technique to update the convective term and crank nicholson scheme for the diffusion term copyright 2007 john wiley sons ltd

complex variable method for eigensolution sensitivity analysis the application of complex variable method cvm for eigenvalue and eigenvector sensitivity analysis is presented gradi ent based methods usually used in structural optimization require accurate sensitivity information cvm is accurate robust and easy to implement as compared to other approximate sensitivity analysis methods by cvm the first order modal sensitivity can b e calculated using an existing structural analysis package this is achieved by making a small complex perturbation to a design variable the resulting eigensolutions would be complex the sensitivity of eigensolutions with respect to that design variable can then be obtained by taking the complex part of the eigensolutions divided by the perturbation since there is no subtraction involved in the sensitivity calculation the approximation is not sensitive to step size thus unlike finite difference method s cvm does not suffer the effect of subtraction cancellation error when a very small step size is used the computation of 1 st order modal sensitivity is not hampered by repeated eigenvalue problem as is the case with other sensitivity analysis methods for repeated eigenvalues cvm automatically provides correct eigenvalue sensitivity and the associated eigenvector sensitivity for a differentiable mode a 5 dof system with an eigenvalue of multiplicity 3 is used to illustrate the salient features of cvm in eigen solution sensitivity analysis nomenclature k stiffness matrix for spring mass system m mass matrix for spring mass system eigenvalue for spring mass system eigenvector for spring mass system derivative of w r t design variable

improving the speed and accuracy of projection type incompressible flow solvers abstract superseding so called first generation incompressible flow solvers of the projection type based on taylor galerkin advection second order pressure damping and element based data structures the current second generation solvers based on high order upwind advection fourth order pressure damping and edge based data structures have now been in use for half a decade and have proven remarkably robust and efficient for many large scale problems in order to achieve higher accuracy and speed these solvers have recently been enhanced in a variety of ways a substepping for advection b implicit treatment of advective terms via sgs and gmres lu sgs iterative solvers c fully implicit time accurate advancement of pressure and velocities and d linelet preconditioning for the pressure poisson equation the combined effect of these third generation improvements leads to speedups of the order of o 1 5 1 10 with similar or even better temporal accuracy as demonstrated on a variety of academic and industrial problems

on the extension of the complex step derivative technique to pseudospectral algorithms the complex step derivative csd technique is a convenient and highly accurate strategy to perform a linearized perturbation analysis to determine a directional derivative via a minor modification of an existing nonlinear simulation code the technique has previously been applied to nonlinear simulation codes such as finite element codes which employ real arithmetic only the present note examines the suitability of this technique for extension to efficient pseudospectral simulation codes which nominally use the fast fourier transform fft to convert back and forth between the physical and transformed representations of the system it is found that if used carefully this extension retains the remarkable accuracy of the csd approach however to perform this extension without sacrificing this accuracy particular care must be exercised specifically the state real and perturbation imaginary components of the complexified system must be transformed separately and arranged in such a manner that they are kept distinct during the process of differentiation in the transformed space in order to avoid the linear combination of the large and small quantities in the analysis it is shown that this is relatively straightforward to implement even in complicated nonlinear simulation codes thus positioning the csd approach as an attractive and relatively simple alternative to hand coding a perturbation a k a tangent linear code for determining the directional derivative even when pseudospectral algorithms are employed

a new algorithm for solving some mechanical problems this paper explores the utility of a discrete singular convolution algorithm for solving certain mechanical problems benchmark mechanical systems including plate vibrations and incompressible flows are employed to illustrate the robustness and to test accuracy of the present algorithm numerical results indicate that the present approach is very accurate efficient and reliable for solving the aforementioned problems

an improved particle correction procedure for the particle level set method the particle level set method d enright r fedkiw j ferziger i mitchell a hybrid particle level set method for improved interface capturing j comput phys 183 2002 83 116 can substantially improve the mass conservation property of the level set method by using lagrangian marker particles to correct the level set function in the under resolved regions in this study the limitations of the particle level set method due to the errors introduced in the particle correction process are analyzed and an improved particle correction procedure is developed based on a new interface reconstruction scheme moreover the zero level set is anchored as the level set functions are reinitialized hence the additional particle correction after the level set reinitialization is avoided with this new scheme a well defined zero level set can be obtained and the disturbances to the interface are significantly reduced consequently the particle reseeding operation will barely result in the loss of interface characteristics and can be applied as frequently as necessary to demonstrate the accuracy and robustness of the proposed method two extreme particle reseeding strategies one without reseeding and the other with reseeding every time step are applied in several benchmark advection tests and the results are compared with each other three interfacial flow cases a 2d surface tension driven oscillating droplet a 2d gas bubble rising in a quiescent liquid and a 3d drop impact onto a liquid pool are simulated to illustrate the advantages of the current method over the level set and the original particle level set methods with regard to the smoothness of geometric properties and mass conservation in real physical applications

analysis of a one dimensional model for the immersed boundary method numerical methods are studied for the one dimensional heat equation with a singular forcing term u t u xx c t delta x alpha t the delta function delta x is replaced by a discrete approximation d h x and the resulting equation is solved by a crank nicolson method on a uniform grid the accuracy of this method is analyzed for various choices of d h the case where c t is specified and also the case where c is determined implicitly by a constraint on the solution at the point a are studied these problems serve as a model for the immersed boundary method of peskin for incompressible flow problems in irregular regions some insight is gained into the accuracy that can be achieved and the importance of choosing appropriate discrete delta functions

uncoupled temporally second order accurate implicit solver of incompressible navier stokes equations an efficient method with temporal second order accuracy for solving the incompressible navier stokes equations in generalized coordinate systems is developed efficiency is obtained by extending an existing fractional step semi implicit solver into an implicit one with implicit approximation of the convection terms a novel three level linearization scheme is proposed to decouple the momentum equations from one another in addition to the fractional step approach that decouples the continuity equation from the momentum equations consequently all of the governing equations are uncoupled without degrading temporal accuracy or stability and without the need for iterations to solve the nonlinear system of equations at each time level the proposed decoupling technique can be used for other systems of nonlinear and time dependent partial differential equations moreover only minor modifications are required to apply the method to existing two level schemes several test cases confirm that the proposed decoupled scheme has temporal second order accuracy and allows the use of large time steps

sensitivity analysis for navier stokes equations on unstructured meshes using complex variables the use of complex variables for determining sensitivity derivatives for turbulent flows is examined although a step size parameter is required the numerical derivatives are not subject to subtractive cancellation errors and therefore exhibit true second order accuracy as the step size is reduced as a result this technique guarantees two additional digits of accuracy each time the step size is reduced one order of magnitude this behavior is in contrast to the use of finite differences which suffer from inaccuracies due to subtractive cancellation errors in addition the complex variable procedure is easily implemented into existing codes

a well conditioned augmented system for solving navier stokes equations in irregular domains an augmented method based on a cartesian grid is proposed for the incompressible navier stokes equations in irregular domains the irregular domain is embedded into a rectangular one so that a fast poisson solver can be utilized in the projection method unlike several methods suggested in the literature that set the force strengths as unknowns which often results in an ill conditioned linear system we set the jump in the normal derivative of the velocity as the augmented variable the new approach improves the condition number of the system for the augmented variable significantly using the immersed interface method we are able to achieve second order accuracy for the velocity numerical results and comparisons to benchmark tests are given to validate the new method a lid driven cavity flow with multiple obstacles and different geometries are also presented

a conservative adaptive projection method for the variable density incompressible navier stokes equations in this paper we present a method for solving the equations governing time dependent variable density incompressible flow in two or three dimensions on an adaptive hierarchy of grids the method is based on a projection formulation in which we first solve advection diffusion equations to predict intermediate velocities and then project these velocities onto a space of approximately divergence free vector fields our treatment of the first step uses a specialized second order upwind method for differencing the nonlinear convection terms that provides a robust treatment of these terms suitable for inviscid and high reynolds number flow density and other scalars are advected in such a way as to maintain conservation if appropriate and free stream preservation our approach to adaptive refinement uses a nested hierarchy of logically rectangular girds with simultaneous refinement of the girds in both space and time the integration algorithm on the grid hierarchy is a recursive procedure in which coarse grids are advanced in time fine grids are advanced multiple steps to reach the same time as the coarse grids and the data at different levels are then synchronized the single grid algorithm is described briefly but the emphasis here is on the time stepping procedure for the adaptive hierarchy numerical examples are presented to demonstrate the algorithms s accuracy and convergence properties and illustrate the behavior of the method an additional example demonstrates the performance of the method on a more realistic problem namely a three dimensional variable density shear layer

derivatives for time spectral computational fluid dynamics using an automatic differentiation adjoint sensitivities for an oscillating onera m6 wing the sensitivities are shown to be accurate to 8 12 digits and the computational cost of the adjoint computations is shown to scale well up to problems of more than 41 million state variables

a coupled level set and volume of fluid method for computing 3d and axisymmetric incompressible two phase flows we present a coupled level set volume of fluid clsvof method for computing 3d and axisymmetric incompressible two phase flows this method combines some of the advantages of the volume of fluid method with the level set method to obtain a method which is generally superior to either method alone we present direct comparisons between computations made with the clsvof method and computations made with the level set method the volume of fluid method and the boundary integral method we also compare our computations to the exact solution for an oscillating ellipse due to lamb and experimental results obtained for a rising gas bubble in liquid obtained by hnat and buckmaster our computational examples focus on flows in which surface tension forces and changes in topology are dominant features of the flow

a diffuse interface model for axisymmetric immiscible two phase flow a diffuse interface model is considered for solving axisymmetric immiscible two phase flow with surface tension the navier stokes ns equations are modified by the addition of a continuum forcing the interface between the two fluids is considered as the half level set of a mass concentration c which is governed by the cahn hilliard ch equation a fourth order degenerate nonlinear parabolic diffusion equation in this work we develop a nonlinear multigrid method to solve the ch equation with degenerate mobility and couple this to a projection method for the incompressible ns equations the diffuse interface method can deal with topological transitions such as breakup and coalescence smoothly without ad hoc cut and connect or other artificial procedures we present results for rayleigh s capillary instability up to forming satellite drops the results agree well with the linear stability theory

a hybrid particle level set method for improved interface capturing in this paper we propose a new numerical method for improving the mass conservation properties of the level set method when the interface is passively advected in a flow field our method uses lagrangian marker particles to rebuild the level set in regions which are underresolved this is often the case for flows undergoing stretching and tearing the overall method maintains a smooth geometrical description of the interface and the implementation simplicity characteristic of the level set method our method compares favorably with volume of fluid methods in the conservation of mass and purely lagrangian schemes for interface resolution the method is presented in three spatial dimensions

survey of shape parameterization techniques for high fidelity multidisciplinary shape optimization a survey is provided of shape parameterization techniques formultidisciplinary optimization and some emerging ideas are highlighted the survey focuses on the suitability of available techniques for multidisciplinary applications of complex cone gurations using high e delity analysis tools such as computational e uid dynamics and computational structural mechanics the suitability criteria are based on the efe ciency effectiveness ease of implementation and availability of analytical sensitivities for geometry and grids a section on sensitivity analysis grid regeneration and grid deformation techniques is also provided