numerical simulation of 3d viscoelastic flows with free surfaces a numerical model is presented for the simulation of viscoelastic flows with complex free surfaces in three space dimensions the mathematical formulation of the model is similar to that of the volume of fluid vof method but the numerical procedures are different a splitting method is used for the time discretization the prediction step consists in solving three advection problems one for the volume fraction of liquid which allows the new liquid domain to be obtained one for the velocity field one for the extra stress the correction step corresponds to solving an oldroyd b fluid flow problem without advection in the new liquid domain two different grids are used for the space discretization the three advection problems are solved on a fixed structured grid made out of small cubic cells using a forward characteristics method the oldroyd b problem without advection is solved using continuous piecewise linear stabilized finite elements on a fixed unstructured mesh of tetrahedrons efficient post processing algorithms enhance the quality of the numerical solution a hierarchical data structure reduces the memory requirements convergence of the numerical method is checked for the pure extensional flow and the filling of a tube numerical results are presented for the stretching of a filament fingering instabilities are obtained when the aspect ratio is large also results pertaining to jet buckling are reported

structural optimization a new dual method using mixed variables a new and powerful mathematical programming method is described which is capable of solving a broad class of structural optimization problems the method employs mixed direct reciprocal design variables in order to get conservative first order approximations to the objective function and to the constraints by this approach the primary optimization problem is replaced with a sequence of explicit subproblems each subproblem being convex and separable it can be efficiently solved by using a dual formulation an attractive feature of the new method lies in its inherent tendency to generate a sequence of steadily improving feasible designs examples of application to real life aerospace structures are offered to demonstrate the power and generality of the approach presented

comparison of dynamic response of structures with uncertain but bounded parameters using non probabilistic interval analysis method and probabilistic approach abstract now by combining the finite element analysis and interval mathematics a new non probabilistic set theoretical models that is interval analysis method is being developed in scientific and engineering communities to predict the variability or uncertainty resulting from the unavoidable scatter in structural parameters and the external excitations as an alternative to the classical probabilistic approaches interval analysis methods of uncertainty were developed for modeling uncertain parameters of structures in which bounds on the magnitude of uncertain parameters are only required not necessarily knowing the probabilistic distribution densities instead of conventional optimization studies where the minimum possible response is sought here an uncertainty modeling is developed as an anti optimization problem of finding the least favorable response and the most favorable response under the constraints within the set theoretical description in this study we presented the non probabilistic interval analysis method for the dynamical response of structures with uncertain but bounded parameters under the condition of the interval vector which contains the uncertain but bounded parameters determined from probabilistic statistical information or stochastic sample test through comparing between the non probabilistic interval analysis method and the probabilistic approach in the mathematical proof and the numerical examples we can see that the region of the dynamical response of structures with uncertain but bounded parameters obtained by the interval analysis method contains that produced by the probabilistic approach in other words the width of the maximum or upper and minimum or lower bounds on the dynamical responses yielded by the probabilistic approach is tighter than those produced by the interval analysis method this kind of results is coincident with the meaning of the probabilistic theory and interval mathematics

a new interface tracking method the polygonal area mapping method we present a new method the polygonal area mapping pam method for tracking a non diffusive immiscible material interface between two materials in two dimensional incompressible flows this method represents material areas explicitly as piecewise polygons traces characteristic points on polygon boundaries along pathlines and calculates new material areas inside interface cells via polygon clippings in a discrete manner the new method has very little spatial numerical diffusion and tracks the interface singularities naturally and accurately in addition to high accuracy the pam method can be directly used on either a structured rectangular mesh or an unstructured mesh without any modifications the mass conservation is enforced by heuristic algorithms adjusting the volume of material polygons the results from a set of widely used benchmark tests show that the pam method is superior to existing volume of fluid vof methods

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

simulation of axisymmetric jets with a finite element navier stokes solver and a multilevel vof approach a multilevel vof approach has been coupled to an accurate finite element navier stokes solver in axisymmetric geometry for the simulation of incompressible liquid jets with high density ratios the representation of the color function over a fine grid has been introduced to reduce the discontinuity of the interface at the cell boundary in the refined grid the automatic breakup and coalescence occur at a spatial scale much smaller than the coarse grid spacing to reduce memory requirements we have implemented on the fine grid a compact storage scheme which memorizes the color function data only in the mixed cells the capillary force is computed by using the laplace beltrami operator and a volumetric approach for the two principal curvatures several simulations of axisymmetric jets have been performed to show the accuracy and robustness of the proposed scheme

embedding problem of fuzzy number space part ii abstract using a concrete structure into which we embed the fuzzy number space e1 several necessary and sufficient conditions of fuzzy set valued functions are given by means of abstract function theory

the point set method front tracking without connectivity abstract we describe a point set method for extracting the normal curvature and surface area from unordered data points residing on a surface this capability relaxes front tracking s reliance on connectivity between interfacial points and allows front tracking to model topological changes at an interface naturally we use this capability in 2d and 3d front tracking calculations to model coalescence we also describe a simple projection scheme which allows us to suppress parasitic currents in front tracking models

interval static displacement analysis for structures with interval parameters in this paper a new method to solve the uncertain static displacement problem of structures with interval parameters is presented it is difficult to obtain all possible solutions with sharp bounds even if an optimum scheme is adopted when there are many uncertain parameters with the interval mathematics the interval finite element equation is developed based on the perturbation and the interval extension the upper and lower bounds of the static displacements are obtained in which the sharp bounds are guaranteed by the interval calculation operators two numerical examples a box cantilever beam and an automobile frame are given to illustrate the validity of the present method copyright 2001 john wiley sons ltd

implicit tracking for multi fluid simulations in this work we introduce a new method for tracking the interfaces among several immiscible fluids to advance the solution in time we use a muscl type scheme which couples a partial volume representation with a level set one to build the numerical fluxes in particular we show the positiveness and conservation properties of this method some numerical tests are given to demonstrate the conservativeness and the performances of our method

modelling merging and fragmentation in multiphase flows with surfer we introduce a new numerical method called surfer for the simulation of two and three dimensional flows with several fluid phases and free interfaces between them we consider incompressible fluids obeying the navier stokes equation with newtonian viscosity in the bulk of each phase capillary forces are taken into account even when interfaces merge or break up fluid interfaces are advanced in time using an exactly volume conserving variant of the volume of fluid algorithm thus allowing for full symmetry between fluid phases the navier stokes equation is solved using staggered finite differences on a mac grid and a split explicit time differencing scheme while incompressibility is enforced using an iterative multigrid poisson solver capillary effects are represented as a stress tensor computed from gradients of the volume fraction function this formulation is completely independent of the topology of interfaces and relatively easy to implement in 3d it also allows exact momentum conservation in the discretized algorithm numerical spurious effects or parasite currents are noticed and compared to similar effects in boltzmann lattice gas methods for immiscible fluids simulations of droplets pairs colliding in 2d and in 3d are shown interface reconnection is performed easily despite the large value of capillary forces during reconnection

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

interface reconstruction with least squares fit and split advection in three dimensional cartesian geometry in this paper we present and analyze different volume of fluid vof reconstruction and advection algorithms that approximate the interface separating two immiscible fluids in the three dimensional space the paper describes the improvement of the reconstruction when a least square fit algorithm which minimizes a distance functional is applied its performance is tested for several smooth surfaces against other simple reconstruction methods then eulerian lagrangian and mixed split advection schemes are presented and analyzed in particular one advection method is discussed that conserves mass exactly for a divergence free velocity field thus allowing computations to machine precision

short note a multi phase flow method with a fast geometry based fluid indicator we present a novel methodology for incompressible multi phase flow simulations in which the fluid indicator is a local signed distance level set function and front tracking is used to evaluate accurately geometric interfacial quantities and forces employing ideas from computational geometry we propose a procedure in which the level set function is obtained at optimal computational cost without having to solve the level set equation and its associated re initialization this new approach is robust and yields an accurate and sharp definition of the distinct bulk phases at all times irrespective of the geometric complexity of the interfaces we illustrate the proposed methodology with an example of surface tension mediated kelvin helmholtz instability

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

numerical simulation of moving contact line problems using a volume of fluid method abstract moving contact lines are implemented in a volume of fluid scheme with piecewise linear interface construction interfacial tension is treated as a continuous body force computed from numerical derivatives of a smoothed volume of fluid function two methods for implementing the contact angle condition are investigated the first extrapolates the volume of fluid function beyond the flow domain on the basis of the condition that its gradient is perpendicular to the interface and that the normal to the interface at the wall is determined by the contact angle the second method treats the problem as a three phase situation and mimics the classical argument of young it is found that the latter approach introduces an artificial localized flow and the extrapolation method is preferable slip is a crucial factor in the spreading of contact lines the numerical method introduces slip at the discrete level effectively introducing a slip length on the order of the mesh size

second order accurate volume of fluid algorithms for tracking material interfaces we introduce two new volume of fluid interface reconstruction algorithms and compare the accuracy of these algorithms to four other widely used volume of fluid interface reconstruction algorithms we find that when the interface is smooth e g continuous with two continuous derivatives the new methods are second order accurate and the other algorithms are first order accurate we propose a design criteria for a volume of fluid interface reconstruction algorithm to be second order accurate namely that it reproduce lines in two space dimensions or planes in three space dimensions exactly we also introduce a second order unsplit volume of fluid advection algorithm that is based on a second order finite difference method for scalar conservation laws due to bell dawson and shubin we test this advection algorithm by modeling several different interface shapes propagating in two simple incompressible flows and compare the results with the standard second order operator split advection algorithm although both methods are second order accurate when the interface is smooth we find that the unsplit algorithm exhibits noticeably better resolution in regions where the interface has discontinuous derivatives such as at corners

an adaptive coupled level set volume of fluid interface capturing method for unstructured triangular grids we present an adaptive coupled level set volume of fluid aclsvof method for interfacial flow simulations on unstructured triangular grids at each time step we evolve both the level set function and the volume fraction the level set function is evolved by solving the level set advection equation using a discontinuous galerkin finite element method the volume fraction advection is performed using a lagrangian eulerian method the interface is reconstructed based on both the level set and the volume fraction information in particular the interface normal vector is calculated from the level set function while the line constant is determined by enforcing mass conservation based on the volume fraction different from previous works we have developed an analytic method for finding the line constant on triangular grids which makes interface reconstruction efficient and conserves volume of fluid exactly the level set function is finally reinitialized to the signed distance to the reconstructed interface since the level set function is continuous the normal vector calculation is easy and accurate compared to a classic volume of fluid method while tracking the volume fraction is essential for enforcing mass conservation the method is also coupled to a finite element based stokes flow solver the code validation shows that our method is second order and mass is conserved very accurately in addition owing to the adaptive grid algorithm we can resolve complex interface changes and interfaces of high curvature efficiently and accurately

a volume of fluid method based on multidimensional advection and spline interface reconstruction a new volume of fluid method for tracking two dimensional interfaces is presented the method involves a multidimensional advection algorithm based on the use of edge matched flux polygons to integrate the volume fraction evolution equation and a spline based reconstruction algorithm the accuracy and efficiency of the proposed method are analyzed using different tests and the results are compared with those obtained recently by other authors despite its simplicity the proposed method represents a significant improvement and compares favorably with other volume of fluid methods as regards the accuracy and efficiency of both the advection and reconstruction steps

metric spaces of fuzzy sets two classes of metrics are introduced for spaces of fuzzy sets their equivalence is discussed and basic properties established a characterisation of compact and locally compact subsets is given in terms of boundedness and p mean equileft continuity and the spaces shown to be locally compact complete and separable metric spaces

a lattice boltzmann approach for free surface flow simulations on non uniform block structured grids in this paper we present a hybrid volume of fluid based algorithm for the simulation of free surface flow problems for the solution of the flow field the lattice boltzmann method is used the additional advection equation for the volume of fluid vof fill level is discretized with a classical finite volume method for the interface reconstruction a piecewise linear interface reconstruction in 3d has been implemented the free surface tracking algorithm is embedded into the 3d non uniform lattice boltzmann based solver virtualfluids freudiger et al 2009 1 freudiger 2009 2 the advection algorithm is verified and validated with well known advection test cases for the validation of the free surface algorithm we present simulations of a breaking dam benchmark

a self adaptive oriented particles level set method for tracking interfaces a new method for tracking evolving interfaces by lagrangian particles in conjunction with a level set approach is introduced this numerical technique is based on the use of time evolution equations for fundamental vector and tensor quantities defined on the front and represents a new and convenient way to couple the advantages of the eulerian description given by a level set function f to the use of lagrangian massless particles the term oriented points out that the information advected by the particles not only concern the spatial location but also the local outward normal vector n to the interface c and the second fundamental tensor the shape operator n the particles are exactly located upon c and provide all the requested information for tracking the interface on their own in addition a self adaptive mechanism suitably modifies at each time step the markers distribution in the numerical domain each particle behaves both as a potential seeder of new markers on c so as to guarantee an accurate reconstruction of the interface and a de seeder to avoid any useless gathering of markers and to limit the computational effort the algorithm is conceived to avoid any transport equation for f and to confine the level set function to the role of a mere post processing tool thus all the numerical diffusion problems usually affecting the level set methodology are removed the method has been tested both on 2d and 3d configurations it carries out a fast reconstruction of the interface and its accuracy is only limited by the spatial resolution of the mesh

a level set method for structural topology optimization this paper presents a new approach to structural topology optimization we represent the structural boundary by a level set model that is embedded in a scalar function of a higher dimension such level set models are flexible in handling complex topological changes and are concise in describing the boundary shape of the structure furthermore a well founded mathematical procedure leads to a numerical algorithm that describes a structural optimization as a sequence of motions of the implicit boundaries converging to an optimum solution and satisfying specified constraints the result is a 3d topology optimization technique that demonstrates outstanding flexibility of handling topological changes fidelity of boundary representation and degree of automation we have implemented the algorithm with the use of several robust and efficient numerical techniques of level set methods the benefit and the advantages of the proposed method are illustrated with several 2d examples that are widely used in the recent literature of topology optimization especially in the homogenization based methods

a surface marker algorithm coupled to an area preserving marker redistribution method for three dimensional interface tracking in this work we present a new interface tracking algorithm based on surface markers to reconstruct and advect interfaces in three dimensional space the algorithm is coupled to a marker redistribution method which is area preserving along the interface the interface is described by a set of closed lines which define a coarse lagrangian quadrangular mesh in the fixed computational grid fixed markers are located where the lines cross and are maintained during the whole simulation at each time step the interface is first reconstructed and then all markers are advected by following streamlines in the reconstruction step new markers are determined by computing the intersections of the interface lines with the grid cell faces and by adding area conservation markers between fixed and grid intersection markers intersection and conservation markers defined at the previous time step are discarded the method maintains a smooth geometrical description of the interface for both two dimensional and three dimensional tests with accurate volume conservation even in very challenging situations where the fluid bodies progressively deform and stretch developing localized regions with very high curvature and thin fluid filaments the method compares favorably with front capturing methods such as volume of fluid and level set and other hybrid techniques such as the particle level set method

a sharp interface method for incompressible two phase flows we present a sharp interface method for computing incompressible immiscible two phase flows it couples the level set and volume of fluid techniques and retains their advantages while overcoming their weaknesses it is stable and robust even for large density and viscosity ratios on the order of 1000 to 1 the numerical method is an extension of the second order method presented by sussman m sussman a second order coupled levelset and volume of fluid method for computing growth and collapse of vapor bubbles journal of computational physics 187 2003 110 136 in which the previous method treated the gas pressure as spatially constant and the present method treats the gas as a second incompressible fluid the new method yields solutions in the zero gas density limit which are comparable in accuracy to the method in which the gas pressure was treated as spatially constant this improvement in accuracy allows one to compute accurate solutions on relatively coarse grids thereby providing a speed up over continuum or ghost fluid methods

reconstructing volume tracking a new algorithm for the volume tracking of interfaces in two dimensions is presented the algorithm is based upon a well defined second order geometric solution of a volume evolution equation the method utilizes local discrete material volume and velocity data to track interfaces of arbitrarily complex topology a linearity preserving piecewise linear interface geometry approximation ensures that solutions generated retain second order spatial accuracy second order temporal accuracy is achieved by virtue of a multidimensional unsplit time integration scheme we detail our geometrically based solution method in which material volume fluxes are computed systematically with a set of simple geometric tasks we then interrogate the method by testing its ability to track interfaces through large controlled topology changes whereby an initially simple interface configuration is subjected to vortical flows numerical results for these strenuous test problems provide evidence for the algorithm s improved solution quality and accuracy

integrals of fuzzy number valued functions abstract using the representation theorem of fuzzy numbers we give an alternative approach for the integrals of fuzzy number valued functions and for the radon nikodym theorem of the fuzzy number valued measures

ergodic theorems for random compact sets and fuzzy variables in banach spaces abstract the individual ergodic theorem is proved in two cases first for compact set valued random variable in a banach space h of type p p 1 second for a random variable having values in the family of all fuzzy subsets of h the set representation of fuzzy subsets is used three types of convergence of fuzzy variables are considered

extreme structural response analysis of truss structures under material uncertainty via linear mixed 0 1 programming this paper concerns the extreme structural response analysis of truss structures taking the possible uncertainties of young s moduli of the bars into consideration the uncertainty of young s modulus is described by a non probability but bounded interval model with the use of this uncertain model it is shown that the structural responses such as nodal displacement and bar stress will attain their extreme values when young s moduli of the bars take either their upper or lower bound value based on this conclusion and simultaneous analysis and design formulation a linear mixed 0 1 programming problem can be constructed to find the extreme values of the structural response by the well established branch and bound approach with global optimality the proposed approach is applied to several test problems for the demonstration of its validity

on henstock integrals of interval valued functions and fuzzy valued functions abstract in this paper the h integrals of interval valued functions and fuzzy valued functions are defined and discussed several necessary and sufficient conditions of h integrability for fuzzy number valued functions are given by means of abstract henstock pettis integral theory

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

cellwise conservative unsplit advection for the volume of fluid method we present a cellwise conservative unsplit ccu advection scheme for the volume of fluid method vof in 2d contrary to other schemes based on explicit calculations of the flux balances the ccu advection adopts a cellwise approach where the pre images of the control volumes are traced backwards through the flow map the donating regions of the fluxes are calculated via the streaklines of the grid intersections represented as polygonal chains whose vertices are determined by backward tracing of particles injected in the flow at different times high order accuracy is obtained from the fourth order runge kutta method where intermediate velocities along pathlines are determined with quadratic temporal and bicubic spatial interpolations the volumes of the donating regions are corrected in order to fulfill the discrete continuity of incompressible flows consequently the calculation produces non overlapping donating regions and pre images with conforming edges to their neighbors resulting in the conservativeness and the boundedness liquid volume fraction inside the interval 0 1 of the ccu advection scheme finally the update of the liquid volume fractions is computed from the intersections of the pre image polygons with the reconstructed interfaces the ccu scheme is tested on several benchmark tests for the vof advection together with the standard piecewise linear interface calculation plic the geometrical errors of the ccu compare favorably with other unsplit vof plic schemes finally potential improvements of the vof method with the use of more precise interface representation techniques and the future extension of the ccu scheme to 3d are discussed cellwise vof update through backward tracing of the cell s pre image accuracy enhanced via bicubic spatial interpolation and polygonal chain streakline the ccu advection scheme is fully conservative and bounded no limitation on the courant number geometrical errors dominated by plic interface representation errors

numerical simulation of drop impact on a liquid liquid interface with a multiple marker front capturing method the gravity driven motion of a droplet impacting on a liquid liquid interface is studied the full navier stokes equations are solved on a fixed uniform grid using a finite difference front capturing method for the representation of fluid fluid interfaces a coupled level set volume of fluid method m sussman e g puckett a coupled level set and volume of fluid method for computing 3d and axisymmetric incompressible two phase flows j comp phys 162 2000 301 337 is used in which we introduce the novel approach of describing separate interfaces with different marker functions as a consequence we prevent numerical coalescence of the droplet and the liquid liquid interface without excessive local grid refinement to validate our method numerical simulations of the drop impact event are compared with experiments z mohamed kassim e k longmire drop impact on a liquid liquid interface phys fluids 15 2003 3263 3273 furthermore a comparison is made with the numerical results of a esmaeeli g tryggvason direct numerical simulations of bubbly flows part 2 moderate reynolds number arrays j fluid mech 385 1999 325 358 for an array of rising bubbles the investigation shows that the multiple marker approach successfully prevents numerical coalescence of interfaces and adequately captures the effect of surface tension

local and global properties of the harmonic polynomial cell method in depth analysis in two dimensions a detailed and systematic analysis is performed on the local and global properties of the recently developed harmonic polynomial cell hpc method a very accurate and efficient field solver for problems governed by the laplace equation at the local cell level a simple rule is identified for the proper choice of harmonic polynomials in the local representation of the velocity potential in cells with symmetry properties the local solution error its convergence rate its dependence on the cell topology its distribution inside the cell and its features across cells with different dimensions are carefully examined with relevant findings for hpc numerical implementations at the global level the error convergence rate is analytically estimated in terms of error contributions from the boundary conditions and from inside the liquid domain in most cases the error associated with boundary conditions dominates the global error in order to minimize it quadtree grid strategies or high order local expressions of the velocity potential are proposed for cells near critical boundary portions to model accurately the boundary conditions on rigid or deformable surfaces with generic geometries 3 different grid strategies are proposed by adopting concepts of immersed boundary method and overlapping grids they are comparatively studied for a circular rigid cylinder in infinite fluid and for the propagation of a free surface wave then an immersed boundary strategy using numerical choices suggested in this paper is successfully compared against a fully nonlinear boundary element method for the case of a surface piercing circular cylinder heaving in otherwise calm water

the supremum and infimum of the set of fuzzy numbers and its application abstract in this paper we prove that the bounded set of fuzzy numbers must exist supremum and infimum and give the concrete representation of supremum and infimum as an application we obtain that the continuous fuzzy valued function on a closed interval exists supremum and infimum and give the precise representation we also show that the bounded fuzzy valued function on a closed interval can define the lower and upper sums and the lower and upper integrals of riemann and riemann stieltjes by the usual way

gradient augmented level set method for phase change simulations abstract a numerical method for the simulation of two phase flow with phase change based on the gradient augmented level set gals strategy is presented sharp capturing of the vaporization process is enabled by i identification of the vapor liquid interface t at the subgrid level ii discontinuous treatment of thermal physical properties except for and iii enforcement of mass momentum and energy jump conditions where the gradients of the dependent variables are obtained at t and are consistent with their analytical expression i e no local averaging is applied treatment of the jump in velocity and pressure at t is achieved using the ghost fluid method the solution of the energy equation employs the sub grid knowledge of t to discretize the temperature laplacian using second order one sided differences i e the numerical stencil completely resides within each respective phase to carefully evaluate the benefits or disadvantages of the gals approach the standard level set method is implemented and compared against the gals predictions the results show the expected trend that interface identification and transport are predicted noticeably better with gals over the standard level set this benefit carries over to the prediction of the laplacian and temperature gradients in the neighborhood of the interface which are directly linked to the calculation of the vaporization rate however when combining the calculation of interface transport and reinitialization with two phase momentum and energy the benefits of gals are to some extent neutralized and the causes for this behavior are identified and analyzed overall the additional computational costs associated with gals are almost the same as those using the standard level set technique

volume of fluid interface tracking with smoothed surface stress methods for three dimensional flows motivated by the need for three dimensional methods for interface calculations that can deal with topology changes we describe a numerical scheme built from a volume of fluid interface tracking technique that uses a piecewise linear interface calculation in each cell momentum balance is computed using explicit finite volume finite differences on a regular cubic grid surface tension is implemented by the continuous surface stress or continuous surface force method examples and verifications of the method are given by comparing simulations to analytical results and experiments for sedimenting droplet arrays and capillary waves at finite reynolds number in the case of a pinching pendant drop both three dimensional and axisymmetric simulations are compared to experiments agreement is found both before and after the reconnections

bounded variation absolute continuity and absolute integrability for fuzzy number valued functions in this paper the concepts of bounded variation and absolute continuity for the fuzzy number valued functions are presented and discussed by means of the representation of the absolute value for fuzzy numbers the relations among bounded variation absolute continuity kaleva s integral and fuzzy henstock integral are characterized especially the representation of absolute continuous fuzzy valued functions is given by henstock kaleva integral of absolutely henstock kaleva integrable fuzzy number valued functions

a new volume of fluid advection algorithm a new vof advection algorithm is presented termed the stream scheme the algorithm uses a linear piecewise method for free surface reconstruction coupled to a unique fully multidimensional method of cell boundary flux integration the performance of the new algorithm is compared against other vof advection algorithms using a variety of standard advection tests in general the performance of the stream scheme appears superior to existing multidimensional and dimensionally split advection algorithms

a mixed markers and volume of fluid method for the reconstruction and advection of interfaces in two phase and free boundary flows in this work we present a new mixed markers and volume of fluid vof algorithm for the reconstruction and advection of interfaces in the two dimensional space the interface is described by using both the volume fraction function c as in vof methods and surface markers which locate the interface within the computational cells the c field and the markers are advected by following the streamlines new markers are determined by computing the intersections of the advected interface with the grid lines then other markers are added inside each cut cell to conserve the volume fraction c a smooth motion of the interface is obtained typical of the marker approach with a good volume conservation as in standard vof methods in this article we consider a few typical two dimensional tests and compare the results of the mixed algorithm with those obtained with vof methods translations rotations and vortex tests are performed showing that many problems of the vof technique can be solved and a good accuracy in the geometrical motion and mass conservation can be achieved

non probabilistic reliability based topology optimization of geometrically nonlinear structures using convex models this paper describes a non probabilistic reliability based topology optimization method for the design of continuum structures undergoing large deformations the variation of the structural system is treated with the multi ellipsoid convex model which is a realistic description of the parameters being inherently uncertain but bounded or lacking sufficient probabilistic data the formulation of the optimal design is established as a volume minimization problem with non probabilistic reliability constraints on the geometrically nonlinear structural behaviour in order to circumvent numerical difficulties in solving the nested double loop optimization problem a performance measure based approach is employed to transform the constraint on the reliability index into one on the concerned performance in conjunction with an efficient adjoint variable scheme for the sensitivity analysis of reliability constraints the optimization problem is solved by gradient based mathematical programming methods three numerical examples for the optimization design of planar structures are presented to illustrate the validity and applicability of the proposed method the obtained optimal solutions show the importance of incorporating various uncertainties in the design problem moreover it is also revealed that the geometrical nonlinearity needs to be accounted for to ensure satisfaction of the reliability constraints in the optimal design of structures with large deformation

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

analytical relations connecting linear interfaces and volume fractions in rectangular grids abstract the computational geometric problems arising when a linear interface cuts a cube are considered they are of interest in particular for the calculation of volume fractions or interface positions in three dimensional interface calculations in the volume of fluid vof methods typically the normal vector is known one then wants to compute the volume fraction knowing the interface position or conversely the interface position knowing the volume fraction explicit expressions of general use are given and the algorithms used to search for solutions are described in detail explicit formulas for cubic roots are found to be less than two thirds as time consuming as newton raphson iterations

forward and inverse structural uncertainty propagations under stochastic variables with arbitrary probability distributions abstract in this study a general frame of the forward and inverse structural uncertainty propagations ups based on the dimension reduction dr method and the derivative lambda probability density function pdf is proposed to handle arbitrary probability distribution for the forward up a dr method is applied to decompose a multivariable system into multiple univariate subsystems and a derivative pdf is adopted to transform the arbitrary probability distribution of each subsystem then the statistical moments and a fitting region are mathematically derived to analyze the fitting ability of the derivative pdf according to whether the kurtosis skewness point lies in or out the fitting region two different strategies combining the gauss gegenbauer quadrature are proposed to implement the forward propagation compared with the conventional methods the proposed method has advantages in higher accuracy stability and efficiency for the inverse propagation because the unknown variable may be arbitrary distribution the general frame based on the derivative pdf and the gauss gegenbauer quadrature are utilized to convert the uncertainty propagation into multiple deterministic calculations based on this optimization method is adopted to accurately obtain the statistical moments and pdfs of the unknown stochastic variables five examples are provided to verify the accuracy and efficiency of the proposed general frame for the forward and inverse ups

parameter perturbation method for dynamic responses of structures with uncertain but bounded parameters based on interval analysis the study was intended to evaluate the range of dynamic responses of structures with uncertain but bounded parameters by using the parameter perturbation method the uncertain parameters were modeled as an interval vector the first order perturbation quantities of responses of the perturbed system were obtained through the parameter perturbation method and then taking advantage of interval mathematics a new algorithm to estimate the response interval was presented comparisons between the parameter perturbation method and the probabilistic approach from mathematical proofs and numerical simulations were performed the numerical results are in agreement with the mathematical proofs the response range given by the parameter perturbation method encloses that obtained by the probabilistic approach the results also show good robustness of the proposed method

connectivity free front tracking method for multiphase flows with free surfaces in this study a connectivity free front tracking method is developed to simulate multiphase flows with free surfaces this method is based on the point set method which does not require any connectivities between interfacial points to represent the interface the main advantage of the connectivity free approach is the easiness in re constructing the interface when large topology change occurs it requires an indicator field to be constructed first based on the existing interface and the surface curvature and normal are then computed using the indicator field here we adopt the reproducing kernel particle method rkpm interpolation function that provides the ability to deal with free surface flows and the flexibility of using non uniform meshes when local fine resolution is needed a points regeneration scheme is developed to construct smooth interfaces and to automatically handle topology changes the mass conservation is verified by performing a single vortex advection test several 2 d and 3 d numerical tests including an oscillating droplet dam breaking two droplet impacting and multi bubble merging are presented to show the accuracy and the robustness of the method

radon nikody m theorem and conditional expectation of fuzzy valued measures and variables abstract the notions of fuzzy valued measure and conditional expectation of a fuzzy valued variable are defined the main result of the paper is a proof of the existence of the radon nikod m derivative of the fuzzy valued measure a convergence theorem for conditional expectations is proved as well

maximal stiffness design of two material structures by topology optimization with nonprobabilistic reliability basedon the multi ellipsoid convex modelandthe quantified measure of the nonprobabilistic reliability topology optimization of two material structures in the presence of parameter uncertainties is investigated the task of the optimal design problem is to distribute a given amount of two candidate materials into the design domain for acquiring the maximal stiffness while satisfying the reliability requirement the extended power law interpolation scheme for material properties is employed for relaxing the two material topological design problem into a continuous valued optimization problem in addition through transforming the minimax type optimization problemintoaseriesofdeterministiconesbyusingasequentialapproximateprogrammingstrategy thispaperaims tomaketheoptimizationdesignnumericallytractable theresultingmathematicalprogrammingproblemsarethen efficientlysolvedbytheassociationofthemethodofmovingasymptoteswithaheuristiciterativemanner numerical investigations reveal that system uncertainties may have considerable effects on the optimal material layout of twomaterial structures the proposed topology optimization methodology could yield a more reasonable two material structuraldesignthan theconventional deterministic counterpart whenthe samereliabilityrequirements need tobe achieved

level set methods an overview and some recent results the level set method was devised by s osher and j a sethian 1988 j comput phys 79 12 49 as a simple and versatile method for computing and analyzing the motion of an interface in two or three dimensions bounds a possibly multiply connected region the goal is to compute and analyze the subsequent motion of under a velocity field v this velocity can depend on position time the geometry of the interface and the external physics the interface is captured for later time as the zero level set of a smooth at least lipschitz continuous function x t i e t x x t 0 is positive inside negative outside and is zero on t topological merging and breaking are well defined and easily performed in this review article we discuss recent variants and extensions including the motion of curves in three dimensions the dynamic surface extension method fast methods for steady state problems diffusion generated motion and the variational level set approach we also give a user s guide to the level set dictionary and technology and couple the method to a wide variety of problems involving external physics such as compressible and incompressible possibly reacting flow stefan problems kinetic crystal growth epitaxial growth of thin films vortex dominated flows and extensions to multiphase motion we conclude with a discussion of applications to computer vision and image processing

fuzzy differential equations abstract this paper deals with fuzzy set valued mappings of a real variable whose values are normal convex upper semicontinuous and compactly supported fuzzy sets in rn we study differentiability and integrability properties of such functions and give an existence and uniqueness theorem for a solution to a fuzzy differential equation

vof evaluation of the surface tension by using variational representation and galerkin interpolation projection abstract in this work we propose a variational approach with cell to point galerkin projections for studying two phase interface advection problems dominated by surface tension a volume of fluid vof algorithm is used for tracking and locating the evolution of the two phase interface on a cartesian grid and a finite element numerical scheme for solving the velocity pressure state the velocity field that drives the evolution of this interface is computed from the weak form of the navier stokes equation where the surface tension force is represented in variational form by the continuous surface force csf and continuous surface stress css methods standard numerical approaches solve the strong form of the navier stokes equations and define the css term by taking the divergence of the surface tension tensor this computation of the divergence term results in a singular force which is difficult to compute when the grid is refined since the tensor is computed in a discontinuous cell by cell way in this work we use the variational formulation of the navier stokes equation and avoid differentiation the tensor which is a function of the unit normal is evaluated over regular sobolev spaces by using a cell to point galerkin projection this allows a regular piece wise continuous representation of the surface tensor and the unit normal based on the vof reconstruction in standard approaches the csf surface force is computed by using the curvature which is the divergence of the unit normal in this paper we recover the curvature with point wise galerkin projection avoiding direct differentiation tests on convergence for two and three dimension in the static and dynamical cases are reported to show the correct representation in the desired spaces this method is also natural for coupling non uniform grid computation of the fluid with cartesian grid of the vof algorithm

a high order projection method for tracking fluid interfaces in variable density incompressible flows we present a numerical method for computing solutions of the incompressible euler or navier stokes equations when a principal feature of the flow is the presence of an interface between two fluids with different fluid properties the method is based on a second order projection method for variable density flows using an approximate projection formulation the boundary between the fluids is tracked with a second order volume of fluid interface tracking algorithm we present results for viscious rayleigh taylor problems at early time with equal and unequal viscosities to demonstrate the convergence of the algorithm we also present computational results for the rayleigh taylor instability in air helium and for bubbles and drops in an air water system without surface tension to demonstrate the behavior of the algorithm on problems with large density and viscosity contrasts

a novel representation of the surface tension force for two phase flow with reduced spurious currents abstract a new numerical model for the solution of the single fluid formulation of the incompressible navier stokes equations for two phase flows is presented the model couples an interface front tracking algorithm to a solver of the navier stokes equations based on the finite element method the interface line is described by an ordered set of markers which are advected along the flow streamlines and from the knowledge of their position the local normal and mean curvature are calculated the local mean curvature has been extended to the whole computational domain and the singular surface tension force has been transformed within the framework of the variational formulation into a regular volume force isoparametric taylor hood elements have been used to discretize the navier stokes equations the resulting system has been solved with a fully multigrid method coupled with a vanka like smoother the model has been tested with a dynamical system characterized by a droplet surrounded by another fluid equilibrium solutions described by the laplace s law are reproduced with great accuracy with a very low level of spurious currents which disappear completely if the exact value of the curvature is used damped oscillations are investigated in terms of the oscillation period the damping coefficient and the critical reynolds number for the transition from periodic to aperiodic regimes good agreement with theoretical solutions and the results of other numerical models has been found when the comparison is applicable

an efficient interface preserving level set redistancing algorithm and its application to interfacial incompressible fluid flow in sussman smereka and osher j comp phys 94 1994 pp 146 159 a numerical scheme was presented for computing incompressible air water flows using the level set method crucial to the above method was a new iteration method for maintaining the level set function as the signed distance from the zero level set in this paper we implement a constraint along with higher order difference schemes in order to make the iteration method more accurate and efficient accuracy is measured in terms of the new computed signed distance function and the original level set function having the same zero level set we apply our redistancing scheme to incompressible flows with noticeably better resolved results at reduced cost we validate our results with experiment and theory we show that our distance level set scheme with the added constraint competes well with available interface tracking schemes for basic advection of an interface we perform basic accuracy checks and more stringent tests involving complicated interfacial structures as with all level set schemes our method is easy to implement

a continuum method for modeling surface tension abstract a new method for modeling surface tension effects on fluid motion has been developed interfaces between fluids of different properties or colors are represented as transition regions of finite thickness across which the color variable varies continuously at each point in the transition region a force density is defined which is proportional to the curvature of the surface of constant color at that point it is normalized so that the conventional description of surface tension on an interface is recovered when the ratio of local transition region thickness to local radius of curvature approaches zero the continuum method eliminates the need for interface reconstruction simplifies the calculation of surface tension enables accurate modeling of two and three dimensional fluid flows driven by surface forces and does not impose any modeling restrictions on the number complexity or dynamic evolution of fluid interfaces having surface tension computational results for two dimensional flows are given to illustrate the properties of the method

on the reinitialization procedure in a narrow band locally refined level set method for interfacial flows a local level set algorithm for simulating interfacial flows described by the two dimensional incompressible navier stokes equations is presented the governing equations are solved using a finite difference discretization on a cartesian grid and a second order approximate projection method the level set transport and reinitialization equations are solved in a narrow band around the interface using an adaptive refined grid which is reconstructed every time step and refined using a simple uniform cell splitting operation within the band instabilities at the border of the narrow band are avoided by smoothing the level set function in the outer part of the band the influence of different pde based reinitialization strategies on the accuracy of the results is investigated the ability of the proposed method to accurately compute interfacial flows is discussed using different tests namely the advection of a circle of fluid in two different time reversed vortex flows the advection of zalesak s rotating disk the propagation of small amplitude gravity and capillary waves at the interface between two superposed viscous fluids in deep water and a classical test of rayleigh taylor instability with and without surface tension effects the interface location error and area loss for some of the results obtained are compared with those of a recent particle level set method copyright 2005 john wiley sons ltd

numerical simulation of free surface incompressible liquid flows surrounded by compressible gas a numerical model for the three dimensional simulation of liquid gas flows with free surfaces is presented the incompressible navier stokes equations are assumed to hold in the liquid domain in the gas domain the velocity is disregarded the pressure is supposed to be constant in each connected component of the gas domain and follows the ideal gas law the gas pressure is imposed as a normal force on the liquid gas interface an implicit splitting scheme is used to decouple the physical phenomena given the gas pressure on the interface the method described in j comput phys 155 1999 439 int j numer meth fluids 42 7 2003 697 is used to track the liquid domain and to compute the velocity and pressure fields in the liquid then the connected components of the gas domain are found using an original numbering algorithm finally the gas pressure is updated from the ideal gas law in each connected component of gas the implementation is validated in the frame of mould filling numerical results in two and three space dimensions show that the effect of pressure in the bubbles of gas trapped by the liquid cannot be neglected

regular article calculation of two phase navier stokes flows using phase field modeling phase field models provide a way to model fluid interfaces as having finite thickness this can allow the computation of interface movement and deformation on fixed grids this paper applies phase field modeling to the computation of two phase incompressible navier stokes flows the navier stokes equations are modified by the addition of the continuum forcing email protected f where c is the composition variable and f is c s chemical potential the equation for interface advection is replaced by a continuum advective diffusion equation with diffusion driven by c s chemical potential gradients the paper discusses how solutions to these equations approach those of the original sharp interface navier stokes equations as the interface thickness e and the diffusivity both go to zero the basic flow physics of phase field interfaces is discussed straining flows can thin or thicken an interface and this must be resisted by a high enough diffusion on the other hand too large a diffusion will overly damp the flow these two constraints result in an upper bound for the diffusivity of o e and a lower bound of o e 2 within these two bounds the phase field navier stokes equations appear to generate an o e error relative to the exact sharp interface equations an o h 2 e 2 numerical method is introduced that is energy conserving in the sense that creation of interface energy by convection is always balanced by an equal decrease in kinetic energy caused by surface tension forcing an o h 4 e 4 compact scheme is introduced that takes advantage of the asymptotic comparatively smooth behavior of the chemical potential for o e accurate phase field models the optimum path to convergence for this scheme appears to be email protected 4 5 the asymptotic rate of convergence corresponding to this is o h 4 5 but results at practical resolutions show that the practical convergence of the method is generally considerably faster than linear extensive analysis and computations show that this scheme is very effective and accurate it allows the accurate calculation of two phase flows with interfaces only two cells wide computational results are given for linear capillary waves and for rayleigh taylor instabilities the first set of computations is compared to exact solutions of the diffuse interface equations and of the original sharp interface equations the rayleigh taylor computations test the ability of the method to compute highly deforming flows these flows include near singular phenomena such as interface coalescences and breakups contact line movement and the formation and breakup of thin wall films grid refinement studies are made and rapid convergence is found for macroscopic flow features such as instability growth rate and propagation speed wavelength and the general physical characteristics of the instability and mass transfer rates

a general frame for uncertainty propagation under multimodally distributed random variables abstract uncertainty propagation under multimodally distributed random variables called multimodal distribution propagation for short is a challenging problem due to the complicate probability density function of the random variables in this paper a general frame based on a new finite mixture model mm constructed by derivative lambda probability density function and polynomial chaos expansion method is put forward to efficiently solve the multimodal distribution propagation problem first mm with high accuracy and extensive applicability is proposed to represent the arbitrary unimodal distributions and multimodal distributions second a pseudo em method proved strictly is proposed to estimate the parameters of this mm third the statistical moments of the response by polynomial chaos expansion method is derived mathematically since the new mixture model can be decomposed into several unimodal probability distributions the multimodal distribution propagation can be successfully converted into several unimodal distribution propagations which further can be easily solved by polynomial chaos expansion method finally the maximum entropy principle is adopted to evaluate the probability density function of the result of every unimodal distribution propagation which is then superimposed into the final probability density function of the system response the proposed frame only requires the first four statistical moments to evaluate the probability density function and avoid calculating the higher statistical moments especially for the situation that the probability distribution of the system response is multimodal four examples are presented to verify the high accuracy and efficiency of the proposed general frame for uncertainty propagation

a novel method of non probabilistic reliability based topology optimization corresponding to continuum structures with unknown but bounded uncertainties abstract a new non probabilistic reliability based topology optimization nrbto method for detailed design of continuum structures is proposed in this paper in which the unknown but bounded ubb uncertainties existing in material and external loads are considered simultaneously based on the interval set theoretic model traditional displacement constraint is transformed into the non probabilistic reliability format in which a novel measuring index namely the optimization feature distance is defined by a combination of the area ratio ideas thus the optimal layout issue is converted by the lightweight design problem with displacement reliability constraints moreover the interval parametric vertex approach and the adjoint vector method are also employed to obtain the sensitivity information between the reliability measurement and design variables by utilizing the method of moving asymptotes mma the optimization problem that is investigated can be solved iteratively two engineering examples are eventually presented to demonstrate the validity and effectiveness of the methodology that has been developed and numerical results show that considering the ubb uncertainty effects during the whole procedure of topology optimization may have a significant influence on the final structural configurations

a front tracking method for viscous incompressible multi fluid flows abstract a method to simulate unsteady multi fluid flows in which a sharp interface or a front separates incompressible fluids of different density and viscosity is described the flow field is discretized by a conservative finite difference approximation on a stationary grid and the interface is explicitly represented by a separate unstructured grid that moves through the stationary grid since the interface deforms continuously it is necessary to restructure its grid as the calculations proceed in addition to keeping the density and viscosity stratification sharp the tracked interface provides a natural way to include surface tension effects both two and three dimensional full numerical simulations of bubble motion are presented

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 plic vof method suited for adaptive moving grids this paper presents a new variant of the volume of fluid vof color function c advection algorithm based on the piecewise linear interface construction plic method suitable for use on general moving grids from several existing methods for reconstructing the linear interface we adopted the least squares volume of fluid interface reconstruction algorithm lvira which can be easily implemented on general grids the distinguishing step in the advection algorithm that takes into account the grid movement is the construction of the donating region containing the fluid passing through corresponding cell faces in a single time step the donating regions are constructed utilizing fluid velocity in cell corners relative to grid corner velocities the method is conservative as it complies with the space conservation law scl and requires a proper definition of the grid velocities and fluxes due to the grid movement the accuracy of the presented advection algorithm is assessed with standard test cases it is comparable with other plic based algorithms on fixed grids while the applicability on adaptive moving grids enables a considerable reduction in the number of grid cells

a front tracking method for the computations of multiphase flow direct numerical simulations of multiphase flows using a front tracking method are presented the method is based on writing one set of governing equations for the whole computational domain and treating the different phases as one fluid with variable material properties interfacial terms are accounted for by adding the appropriate sources as functions at the boundary separating the phases the unsteady navier stokes equations are solved by a conventional finite volume method on a fixed structured grid and the interface or front is tracked explicitly by connected marker points interfacial source terms such as surface tension are computed on the front and transferred to the fixed grid advection of fluid properties such as density is done by following the motion of the front the method has been implemented for fully three dimensional flows as well as for two dimensional and axisymmetric ones first the method is described for the flow of two or more isothermal phases the representation of the moving interface and its dynamic restructuring as well as the transfer of information between the moving front and the fixed grid are discussed applications and extensions of the method to homogeneous bubbly flows atomization flows with variable surface tension solidification and boiling are then presented

the method of moving asymptotes a new method for structural optimization a new method for non linear programming in general and structural optimization in particular is presented in each step of the iterative process a strictly convex approximating subproblem is generated and solved the generation of these subproblems is controlled by so called moving asymptotes which may both stabilize and speed up the convergence of the general process

safety margin analysis of buckling for structures with unknown but bounded uncertainties abstract a safety margin of buckling for structures with unknown but bounded uncertainties is proposed in this paper the perturbation matrixes consisting of the elastic stiffness matrix and the geometric stiffness matrix are introduced to derive the concept of safety margin the concept of safety margin of buckling is first proposed to quantify the influence of uncertain parameters on structural buckling based on the spectrum norm and the singular value analysis the mathematical basis of the safety margin is expounded furthermore to obtain a more accurate estimation of safety margin the correction factor for the safety margin is introduced through the interval extension of interval analysis two engineering examples compared with traditional interval analysis method are utilized to demonstrate the validity of the proposed safety margin and numerical results indicate that the safety margin can provide a relatively accurate index to indicate the allowable perturbation range of the perturbation matrix of a structure under given loads

fuzzy random variables abstract in this paper we define the concepts of fuzzy random variable and the expectation of a fuzzy random variable the new definition of expectation generalizes the integral of a set valued function we derive some properties of these new concepts by considering a suitable generalization of the hausdorff metric we derive the lebesgue dominated convergence type theorem

a geometrical predictor corrector advection scheme and its application to the volume fraction function we present a multidimensional eulerian advection method for interfacial and incompressible flows in two dimensional cartesian geometry in the scheme we advect the grid nodes backwards along the streamlines to compute the pre images of the grid lines these pre images are approximated by continuous piecewise linear lines the enforcement of the discrete version of the incompressibility constraint is a very important issue to determine correctly the flux polygons and to reduce considerably the integration discretization and interpolation numerical errors the proposed method compares favorably with other previous multidimensional advection methods as long as the initial interface line is well reconstructed conversely we show that when the interface is very fragmented the total numerical error is completely dominated by the reconstruction error and in these conditions it is very difficult to assess which advection scheme is the most reliable one

an improved plic vof method for tracking thin fluid structures in incompressible two phase flows a drawback common to plic schemes is the difficulty of reconstructing the interface when filaments of a thickness smaller than the cell size are present in this work we present an improvement of the reconstruction method proposed by lopez et al a volume of fluid method based on multidimensional advection and spline interface reconstruction j comput phys 195 2004 718 for two dimensional flows which allows tracking fluid structures thinner than the cell size by allowing the interface to be represented in each cell by two non contiguous linear segments the method is based on using markers that are placed every time step at the mid points of the reconstructed cell interface segments the markers are used to improve the orientation and location of the interface segments although the method essentially remains of a vof type numerical tests show that a substantial improvement in accuracy is achieved over previous volume of fluid methods and that the proposed method compares well even with a recent hybrid markers and volume of fluid method

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

an oriented particle level set method based on surface coordinates in this work we present a new method for tracking evolving interfaces like the adaptive oriented particle level set method aopls ianniello and mascio 2010 it uses a combination of unconnected particles containing geometric information and a level set function to track the interface in the present method the geometric information solved for in each particle is a set of tangent vectors and the corresponding curvature tensor of the interface compared to the aopls method the amount of geometric data stored in each particle is roughly halved while still retaining a description of the interface that includes both orientation and curvature a mechanism for ensuring a sufficiently dense distribution of particles on the interface is also included from the information stored in each particle a continuous representation of the interface is obtained by transferring the information to a level set function stored on an eulerian grid unlike the aopls method the present method takes the curvature into account when the zero level set is constructed this results in significant improvements in regions of high curvature two different formulations of the new method have been explored in the first the tangent vectors deform purely according to the velocity field in the second the tangent vectors are forced to coincide with the principal directions of the surface both formulations are shown to be applicable