optimality conditions for shape and topology optimization subject to a cone constraint this paper provides first order necessary optimality conditions for simultaneous shape and topology optimization subject to a cone constraint these conditions are expressed with the help of the shape and topological derivatives of the objective and constraint functionals several examples of applications are given

ellam for resolving the kinematics of two dimensional resistive magnetohydrodynamic flows we combine the finite element method with the eulerian lagrangian localized adjoint method ellam to solve the convection diffusion equations that describe the kinematics of magnetohydrodynamic flows i e the advection and diffusion of a magnetic field simulations of three two dimensional test problems are presented and in each case we analyze the energy of the magnetic field as it evolves towards its equilibrium state our numerical results highlight the accuracy and efficiency of the ellam approach for convection dominated problems

topological sensitivity analysis for elliptic differential operators of order 2m abstract the topological derivative is defined as the first term of the asymptotic expansion of a given shape functional with respect to a small parameter that measures the size of a singular domain perturbation it has applications in many different fields such as shape and topology optimization inverse problems image processing and mechanical modeling including synthesis and or optimal design of microstructures fracture mechanics sensitivity analysis and damage evolution modeling the topological derivative has been fully developed for a wide range of second order differential operators in this paper we deal with the topological asymptotic expansion of a class of shape functionals associated with elliptic differential operators of order 2m m 1 the general structure of the polarization tensor is derived and the concept of degenerate polarization tensor is introduced we provide full mathematical justifications for the derived formulas including precise estimates of remainders

third order iterative methods with applications to hammerstein equations a unified approach the geometrical interpretation of a family of higher order iterative methods for solving nonlinear scalar equations was presented in s amat s busquier j m gutierrez geometric constructions of iterative functions to solve nonlinear equations j comput appl math 157 1 2003 197 205 this family includes as particular cases some of the most famous third order iterative methods chebyshev methods halley methods super halley methods c methods and newton type two step methods the aim of the present paper is to analyze the convergence of this family for equations defined between two banach spaces by using a technique developed in j a ezquerro m a hernandez halley s method for operators with unbounded second derivative appl numer math 57 3 2007 354 360 this technique allows us to obtain a general semilocal convergence result for these methods where the usual conditions on the second derivative are relaxed on the other hand the main practical difficulty related to the classical third order iterative methods is the evaluation of bilinear operators typically second order frechet derivatives however in some cases the second derivative is easy to evaluate a clear example is provided by the approximation of hammerstein equations where it is diagonal by blocks we finish the paper by applying our methods to some nonlinear integral equations of this type

a finite volume eulerian lagrangian localized adjoint method for solution of the contaminant transport equations in two dimensional multiphase flow systems an eulerian lagrangian localized adjoint method ellam numerical solution is developed for the multiphase contaminant transport equations in two dimensions the ellam uses finite volume test functions in the space time domain defined by the characteristics of the hyperbolic part of the governing equation the use of the characteristics results in an approximation that allows large time steps while still maintaining accurate solutions this greatly reduces the computational effort required to find solutions to the advection dispersion equation combination of the finite volume test functions with the conservative form of the governing equation results in a local conservation of mass property for problems with a constant saturation such as that of contaminant transport in the saturated zone this property is highly advantageous as the resulting numerical approximation conserves mass both globally and locally for problems with a variable saturation the requirement of local conservation of mass is too stringent particularly around first type boundaries where oscillations can occur because of errors inherent in the numerical determination of fluid velocities and in the backtracking routine a combined conservative nonconservative ellam is developed with an ellam formulation based on the nonconservative form of the governing equation being applied to subdomains intersecting first type boundaries and a conservative ellam being used for all other subdomains the combined conservative nonconservative ellam is compared to a galerkin finite element scheme and is found to have greatly superior performance requiring far fewer time steps to obtain a solution of equivalent accuracy

an ellam scheme for advection diffusion equations in two dimensions we develop an eulerian lagrangian localized adjoint method ellam to solve two dimensional advection diffusion equations with all combinations of inflow and outflow dirichlet neumann and flux boundary conditions the ellam formalism provides a systematic framework for implementation of general boundary conditions leading to mass conservative numerical schemes the computational advantages of the ellam approximation have been demonstrated for a number of one dimensional transport systems practical implementations of ellam schemes in multiple spatial dimensions that require careful algorithm development are discussed in detail in this paper extensive numerical results are presented to compare the ellam scheme with many widely used numerical methods and to demonstrate the strength of the ellam scheme

computation of high order derivatives in optimal shape design in shape optimization problems each computation of the cost function by the finite element method leads to an expensive analysis the use of the second order derivative can help to reduce the number of analyses fujii 4 10 was the first to study this problem j simon 19 gave the second order derivative for the navier stokes problem and the authors describe in 8 11 a method which gives an intrinsic expression of the first and second order derivatives on the boundary of the involved domain in this paper we study higher order derivatives but one can ask the following questions are they expensive to calculate are they complicated to use are they imprecise are they useless medskip noindent at first sight the answer seems to be positive but classical results of v strassen 20 and j morgenstern 13 tell us that the higher order derivatives are not expensive to calculate and can be computed automatically the purpose of this paper is to give an answer to the third question by proving that the higher order derivatives of a function can be computed with the same precision as the function itself we prove also that the derivatives so computed are equal to the derivatives of the discrete problem see diagram 1 we call the discrete problem the finite dimensional problem processed by the computer this result allows the use of automatic differentiation 5 6 which works only on discrete problems furthermore the computations of taylor s expansions which are proposed at the end of this paper could be a partial answer to the last question

sensitivity analysis in kinematically incompatible models abstract a natural question that arises when working with the so called kinematically incompatible models is about the positioning of the internal artificial boundary that splits the domain of analysis into the sub domains for which the different incompatible models are established although the experience in modeling might provide an ad hoc solution to such matter it may also draw misleading conclusions here the sensitivity analysis furnishes a systematic and reliable framework to study the impact of the positioning of the artificial boundary over the solution of the problem in this sense the present work is concerned with the sensitivity analysis of the variational formulation corresponding to the kinematically incompatible models when the boundary over which the incompatibility between the models takes place is changed assuming the presence of a discontinuity in the fields over a given interface between two incompatible models this analysis allows to measure the sensitivity of a given cost functional when the location of such interface is changed the application for which this analysis is envisaged is to assess the correctness or incorrectness in the definition of the positioning of such coupling interface between dimensionally heterogeneous domains also some numerical results are provided to give numerical evidence of the usefulness of the present tool

higher order topological derivatives in elasticity the topological derivative provides the variation of a response functional when an infinitesimal hole of a particular shape is introduced into the domain in this work we compute higher order topological derivatives for elasticity problems so that we are able to obtain better estimates of the response when holes of finite sizes are introduced in the domain a critical element of our algorithm involves the asymptotic approximation for the stress on the hole boundary when the hole size approaches zero it consists of a composite expansion that is based on the responses of elasticity problems on the domain without the hole and on a domain consisting of a hole in an infinite space we present a simple example in which the higher order topological derivatives of the total potential energy are obtained analytically and by using the proposed asymptotic expansion we also use the finite element method to verify the topological asymptotic expansion when the analytical solution is unknown

fast upwind and eulerian lagrangian control volume schemes for time dependent directional space fractional advection dispersion equations abstract we develop control volume methods for two dimensional time dependent advection dominated directional space fractional advection dispersion equations with the directional space fractional derivative weighted in all the directions by a probability measure in the unit circle which are used to model the anisotropic superdiffusive transport of solutes in groundwater moving through subsurface heterogeneous porous media we develop a fast upwind control volume method for the governing equation to eliminate the spurious numerical oscillations that often occur in space centered numerical discretizations of advection term which are relatively straightforward to implement we also develop a eulerian lagrangian control volume method for the governing equation which symmetrizes the governing equation by combining the time derivative term and the advection term into a material derivative term along characteristic curves both methods are locally mass conservative which are essential in these applications due to the nonlocal nature of the directional space fractional differential operators corresponding numerical discretizations usually generate full stiffness matrices conventional direct solvers tend to require o n 2 memory requirement and have o n 3 computational complexity per time step where n is the number of spatial unknowns which is computationally significantly more expensive than the numerical approximations of integer order advection diffusion equations based on the analysis of the structure of stiffness matrix we propose a fast krylov subspace iterative solver to accelerate the numerical approximations of both the upwind and eulerian lagrangian control volume methods which reduce computational complexity from o n 3 by a direct solver to o n log n per krylov subspace iteration per time step and a memory requirement from o n 2 to o n numerical results are presented to show the utility of the methods

a staggered approach to shape and topology optimization using the traction method and an evolutionary type advancing front algorithm abstract we investigate a novel approach for structural shape optimization on the basis of complementary shape and topological sensitivity analysis as in early approaches to shape optimization the domain variation is specified by modification of boundary nodal points hence leading to an updated lagrangian description for the course of optimization to overcome the formation of oscillating boundaries in the optimal design trials we employ the traction method to establish smooth descent directions for shape variation therein an auxiliary elastic problem is solved in which the shape sensitivity constitutes the external loading and the deformation of the auxiliary elastic body is used as a descent direction for shape variation rather than the shape sensitivity itself we complement this method through an evolutionary type element removal procedure that is based on the topological sensitivity such that an advancing front algorithm is gradually removing elements from the design boundary of the domain once the minimum topological sensitivity is no longer encountered at the design boundary we create a hole in the domain again using the topological sensitivity to specify its exact location and resume the element removal procedure for the newly established design boundary since this approach yields only a vague estimate of the true optimal shape of newly established holes the traction method is again used for shape variation of the extended design boundary

performance analysis of multi frequency topological derivative for reconstructing perfectly conducting cracks abstract this paper concerns a fast one step iterative technique of imaging extended perfectly conducting cracks with dirichlet boundary condition in order to reconstruct the shape of cracks from scattered field data measured at the boundary we introduce a topological derivative based electromagnetic imaging functional operated at several nonzero frequencies the structure of the imaging functionals is carefully analyzed by establishing relationships with infinite series of bessel functions for the configurations of both symmetric and non symmetric incident field directions identified structure explains why the application of incident fields with symmetric direction operated at multiple frequencies guarantees a successful reconstruction various numerical simulations with noise corrupted data are conducted to assess the performance effectiveness robustness and limitations of the proposed technique

the topological asymptotic for pde systems the elasticity case the aim of the topological sensitivity analysis is to obtain an asymptotic expansion of a design functional with respect to the creation of a small hole in this paper such an expansion is obtained and analyzed in the context of linear elasticity for general functionals and arbitrary shaped holes by using an adaptation of the adjoint method and a domain truncation technique the method is general and can be easily adapted to other linear pdes and other types of boundary conditions

on the finite volume element method the finite volume element method fve is a discretization technique for partial differential equations it uses a volume integral formulation of the problem with a finite partitioning set of volumes to discretize the equations then restricts the admissible functions to a finite element space to discretize the solution this paper develops discretization error estimates for general selfadjoint elliptic boundary value problems with fve based on triangulations with linear finite element spaces and a general type of control volume we establisho h estimates of the error in a discreteh 1 semi norm under an additional assumption of local uniformity of the triangulation the estimate is improved too h 2 results on the effects of numerical integration are also included

a two grid method for incompressible miscible displacement problems by mixed finite element and eulerian lagrangian localized adjoint methods abstract in this paper we present a scheme for solving two dimensional miscible displacement problems using eulerian lagrangian localized adjoint methods and mixed finite element methods since only the velocity and not the pressure appears explicitly in the concentration equation an eulerian lagrangian localized adjoint method is used to solve the concentration equation and a mixed finite element method is used for the pressure equation to linearize and decouple the mixed method equations we use a two grid algorithm based on the newton iteration method for this fully discrete problems first we solve the original nonlinear equations on the coarse grid then we solve the linearized problem on the fine grid using newton iteration once it is shown that the coarse grid can be much coarser than the fine grid and achieve asymptotically optimal approximation as long as the mesh sizes satisfy h o h 1 2 in this paper

a level set method for shape and topology optimization of both structure and support of continuum structures abstract we present a level set method for the shape and topology optimization of both structure and support two level set functions are used to implicitly represent a structure the traction free boundary and the dirichlet boundary are represented separately and are allowed to be continuously changed during the optimization the optimization problem of minimum compliance is considered the shape derivatives of both boundaries are derived by using a lagrangian function and the adjoint method the finite element analysis is done by modifying a fixed background mesh and we do not use the artificial weak material numerical examples in two dimensions are investigated

solving nonlinear integral equations of fredholm type with high order iterative methods the application of high order iterative methods for solving nonlinear integral equations is not usual in mathematics but in this paper we show that high order iterative methods can be used to solve a special case of nonlinear integral equations of fredholm type and second kind in particular those that have the property of the second derivative of the corresponding operator have associated with them a vector of diagonal matrices once a process of discretization has been done

a consistent relaxation of optimal design problems for coupling shape and topological derivatives in this article we introduce and analyze a general procedure for approximating a black and white shape and topology optimization problem with a density optimization problem allowing for the presence of grayscale regions our construction relies on a regularizing operator for smearing the characteristic functions involved in the exact optimization problem and on an interpolation scheme which endows the intermediate density regions with fictitious material properties under mild hypotheses on the smoothing operator and on the interpolation scheme we prove that the features of the approximate density optimization problem material properties objective function etc converge to their exact counterparts as the smoothing parameter vanishes in particular the gradient of the approximate objective functional with respect to the density function converges to either the shape or the topological derivative of the exact objective these results shed new light on the connections between these two different notions of sensitivities for functions of the domain and they give rise to different numerical algorithms which are illustrated by several experiments

a linear backward euler scheme for the saturation equation regularity results and consistency we consider a linearization of a numerical scheme for the saturation equation or porous medium equation email protected email protected email protected f s email protected email protected k s s 0 through first order expansions of the fractional function f and the inverse of the function k s email protected 0 sk t email protected after a regularization of the porous medium equation we establish a regularity result for the continuous galerkin method and a regularity result for the linearized scheme analogous to the corresponding nonlinear scheme we then show that the linearized scheme is consistent with the nonlinear scheme analyzed in a previous work

topological derivative based topology optimization of structures subject to drucker prager stress constraints abstract an algorithm for topology optimization of elastic structures under plane stress subject to the drucker prager stress constraint is presented the algorithm is based on the use of the topological derivative of the associated objective functional in conjunction with a level set representation of the structure domain in this context a penalty functional is proposed to enforce the point wise stress constraint and a closed formula for its topological derivative is derived the resulting algorithm is of remarkably simple computational implementation it does not require post processing procedures of any kind and features only a minimal number of user defined algorithmic parameters this is in sharp contrast with current procedures for topological structural optimization with local stress constraints the effectiveness and efficiency of the algorithm presented here are demonstrated by means of numerical examples the examples show in particular that it can easily handle structural optimization problems with underlying materials featuring strong asymmetry in their tensile and compressive yield strengths

cad inspired hierarchical partition of unity constructions for nurbs based meshless design analysis and optimization in this paper we develop a unified representational paradigm for design and analysis that is inspired by the set theoretic boolean operations of the constructive solid geometry csg procedure of computer aided design cad we develop the notion of a primitive design state corresponding to a csg primitive region which is characterized by a description of the geometrical shape of the primitive and by a description of the material distribution within the primitive we analogously define a primitive behaviour state that is associated with a primitive design state and that is determined through a global analysis problem we define a global multi level design problem to determine the primitive design states we define boolean operations on the fields belonging to primitive design and behaviour spaces and show that the compositions of these fields amount to a hierarchical partition of unity construction we propose to use non uniform rational b splines nurbs to discretize the geometry material and behavioural fields local approximations defined on the primitive regions we show that this leads to recursive partition of unity constructions on the knot spaces sub local approximations due to the partition of unity property of nurbs basis functions since the primitive shapes are parametrically defined using nurbs the numerical quadrature is carried out on the parametric sub regions of the nurbs entities this eliminates the need for either a background mesh or a pseudo mesh for quadrature the developed methodology is implemented in a symbolic meshless computational framework written using the java language jnurbs and was demonstrated on two classes of problems the first class is the simultaneous optimal design of shape and multi material distributions the procedure is argued as being a generalization of classical topology optimization to designing objects made of functionally graded materials with pre defined geometrical constraints such as holes or fixed boundary shape the second class of problems demonstrated in the paper is optimal design to mitigate the effects of cracks here we identify the optimal locations of holes introduced to reduce the stress intensity factor of an edge crack copyright 2007 john wiley sons ltd

isogeometric topology optimization using trimmed spline surfaces in the present work a new spline based topology optimization using trimmed spline surfaces and the isogeometric analysis is proposed in the proposed approach the trimmed surface analysis which can treat topologically complex spline surfaces using trimming information provided by cad systems is employed for structural response analysis and sensitivity calculation in the topology optimization the outer and inner boundaries of design models are represented by a spline surface and trimming curves design variables used in this approach are the coordinates of control points of a spline surface and those of trimming curves new sensitivity formulations for the control points in the trimmed surface analysis are proposed and their efficiency and accuracy are verified the creation of new inner fronts during optimization is allowed for the topological flexibility an inner front merging algorithm is also presented the proposed spline based topology optimization is used to solve some benchmarking problems design space dependency which is one of serious shortcomings in conventional topology optimization approaches is naturally eliminated by the proposed spline based optimization design dependent load problems which are difficult to treat with conventional grid based topology optimization methods are easily dealt with by the proposed one it is also shown that post processing effort for converting to cad model is eliminated by using the same spline information in numerical analysis and design optimization

a conservative characteristic finite volume element method for solution of the advection diffusion equation abstract we develop a modified characteristic finite volume element method for two dimensional advection diffusion problem in a convex spacial domain using quasi uniform triangular discretization and piecewise linear element this scheme conserves mass globally and locally on the lagrangian space time control volume we give the optimal order h1 norm and optimal l2 norm error estimates finally we present some numerical examples

a characteristics mixed covolume method for a convection dominated transport problem in this paper we propose a characteristics mixed covolume method for approximating the solution to a convection dominated transport problem the method is a combination of characteristic approximation to handle the convection term in time and mixed covolume method spatial approximation to deal with the diffusion term the velocity and press are approximated by the lowest order raviart thomas mixed finite element space on rectangles the projection of a mixed covolume element is introduced we prove its first order optimal rate of convergence for the approximate velocities in the l 2 norm as well as for the approximate pressures in the l 2 norm

an optimal order error estimate for a family of ellam mfem approximations to porous medium flow mathematical models used to describe porous medium flow lead to coupled systems of time dependent nonlinear partial differential equations which present serious mathematical and numerical difficulties standard methods tend to generate numerical solutions with nonphysical oscillations or numerical dispersion along with spurious grid orientation effect the ellam mfem time stepping procedure in which an eulerian lagrangian localized adjoint method ellam is used to solve the transport equation and a mixed finite element method mfem is used for the pressure equation simulates porous medium flow accurately even if large spatial grids and time steps are used in this paper we prove an optimal order error estimate for a family of ellam mfem approximations

a fractional step θ method for convection diffusion problems abstract in this article we analyze the fractional step method for the time dependent convection diffusion equation in our implementation we completely separate the convection operator from the diffusion operator and stabilize the convective problem using a streamline upwinded petrov galerkin supg method we establish a priori error estimates and show that the optimal value of yields a scheme that is second order in time numerical computations are presented which demonstrate the method and support the theoretical results

streamline upwind petrov galerkin formulations for convection dominated flows with particular emphasis on the incompressible navier stokes equations abstract a new finite element formulation for convection dominated flows is developed the basis of the formulation is the streamline upwind concept which provides an accurate multidimensional generalization of optimal one dimensional upwind schemes when implemented as a consistent petrov galerkin weighted residual method it is shown that the new formulation is not subject to the artificial diffusion criticisms associated with many classical upwind methods the accuracy of the streamline upwind petrov galerkin formulation for the linear advection diffusion equation is demonstrated on several numerical examples the formulation is extended to the incompressible navier stokes equations an efficient implicit pressure explicit velocity transient algorithm is developed which accomodates several treatments of the incompressibility constraint and allows for multiple iterations within a time step the effectiveness of the algorithm is demonstrated on the problem of vortex shedding from a circular cylinder at a reynolds number of 100

topology optimization with pressure load through a level set method abstract the topology optimization problem with pressure load is solved by using a level set method the free boundary and the pressure boundary of a structure are represented separately as two zero level sets of two level set functions and they are independently propagated during the optimization by solving two hamilton jacobi equations in order to prevent the two boundaries from touching or crossing each other the design velocities of the two boundaries that amount to the steepest descent directions are modified the optimization problem of minimum compliance with perimeter regularization is considered the shape derivatives of the two boundaries are derived by using the material derivative approach and the adjoint method the finite element analysis is done through an eulerian method by employing a fixed mesh and an artificial weak material that represents void numerical examples in two dimensions are investigated

application of iterative processes of r order at least three to operators with unbounded second derivative abstract in this paper we apply a family of newton like methods which contains the best known iterative processes to operator equations where the usual convergence conditions are relaxed we weaken these conditions by assuming f x0 and f x f y x y with a non decreasing continuous real function our results include the ones obtained when the convergence of the family is studied under lipschitz continuous or holder continuous conditions for the second derivative of the operator involved to finish we apply the study to boundary value problems

a note on the halley method in banach spaces abstract we provide a convergence theorem under the usual ostrowski kantorovich conditions for an equivalent form of the halley method in banach spaces the best possible upper error bound for the method is obtained some applications to the solution of quadratic operator equations are provided

stability and resolution analysis for a topological derivative based imaging functional the aim of this paper is to study a topological derivative based anomaly detection algorithm we compare its performance with other imaging approaches such as multiple signal classification backpropagation and kirchhoff migration we also investigate its stability with respect to medium and measurement noises as well as its resolution a simple postprocessing of the data set is introduced and shown to be essential in order to obtain an efficient topological based imaging functional both in terms of resolution and stability

explicit level set based topology optimization using an exact heaviside function and consistent sensitivity analysis summary this paper presents a level set based topology optimization method based on numerically consistent sensitivity analysis the proposed method uses a direct steepest descent update of the design variables in a level set method the level set nodal values an exact heaviside formulation is used to relate the level set function to element densities the level set function is not required to be a signed distance function and reinitialization is not necessary using this approach level set based topology optimization problems can be solved consistently and multiple constraints treated simultaneously the proposed method leads to more insight in the nature of level set based topology optimization problems the level set based design parametrization can describe gray areas and numerical hinges consistency causes results to contain these numerical artifacts we demonstrate that alternative parameterizations level set based or density based regularization can be used to avoid artifacts in the final results the effectiveness of the proposed method is demonstrated using several benchmark problems the capability to treat multiple constraints shows the potential of the method furthermore due to the consistency the optimizer can run into local minima a fundamental difficulty of level set based topology optimization more advanced optimization strategies and more efficient optimizers may increase the performance in the future copyright 2012 john wiley sons ltd

a singularity avoiding moving least squares scheme for two dimensional unstructured meshes moving least squares interpolation schemes are in widespread use as a tool for numerical analysis on scattered data in particular they are often employed when solving partial differential equations on unstructured meshes which are typically needed when the geometry defining the domain is complex it is known that such schemes can be singular if the data points in the stencil happen to be in certain special geometric arrangements however little research has specifically addressed this issue in this paper a moving least squares scheme is presented which is an appropriate tool for use when solving partial differential equations in two dimensions and the precise conditions under which singularities occur are identified the theory is used to develop a stencil building algorithm which automatically detects singular stencils and corrects them in an efficient manner while attempting to maintain stencil symmetry as closely as possible finally the scheme is applied in a convection diffusion equation solver and an incompressible navier stokes solver and the results are shown to compare favourably with known analytical solutions and previously published results

cfl condition and boundary conditions for dgm approximation of convection diffusion we propose a general method for the design of discontinuous galerkin methods dgms for nonstationary linear equations the method is based on a particular splitting of the bilinear forms that appear in the weak dgm we prove that an appropriate time splitting gives a stable linear explicit scheme whatever the order of the polynomial approximation numerical results are presented

second order topological sensitivity analysis abstract the topological derivative provides the sensitivity of a given cost function with respect to the insertion of a hole at an arbitrary point of the domain classically this derivative comes from the second term of the topological asymptotic expansion dealing only with infinitesimal holes however for practical applications we need to insert holes of finite size therefore we consider one more term in the expansion which is defined as the second order topological derivative in order to present these ideas in this work we apply the topological shape sensitivity method as a systematic approach to calculate first as well as second order topological derivative for the poisson s equations taking the total potential energy as cost function and the state equation as constraint furthermore we also study the effects of different boundary conditions on the hole neumann and dirichlet both homogeneous finally we present some numerical experiments showing the influence of the second order topological derivative in the topological asymptotic expansion which has two main features it allows us to deal with hole of finite size and provides a better descent direction in optimization process

an optimal order estimate for eulerian lagrangian localized adjoint methods for variable coefficient advection reaction problems previously we developed eulerian lagrangian localized adjoint methods ellams to solve advection reaction problems ellams provide a systematic approach to treat boundary conditions and to conserve mass and have proven to be powerful methods for advection dominated problems in this paper the authors conduct theoretical analysis for these ellams and prove that they have an optimal order convergence rate moreover the estimates involve only the derivatives of the exact solution in space and along the characteristics in contrast many existing methods have only suboptimal order error estimates for advection reaction problems and the estimates involve the temporal derivatives of the exact solution which are usually much larger than the derivatives along the characteristics key words advection reaction equations eulerian lagrangian method convergence analysis optimal order error estimates

on a characterization of some newton like methods of r order at least three in this paper we extend to banach spaces the result given by gander in amer math monthly 92 1985 131 to obtain a characterization of newton like iterative process with r order of convergence at least three to do this we consider the construction and the study of the semilocal convergence of a multiparametric family of iterative processes in banach spaces for solving the nonlinear equation f x 0

adaptive finite element methods for conservation laws based on a posteriori error estimates we prove a posteriori error estimates for a finite element method for systems of strictly hyperbolic conservation laws in one space dimension and design corresponding adaptive methods the proof of the a posteriori error estimates is based on a strong stability estimate for an associated dual problem together with the galerkin orthogonality of the finite element method the strong stability estimate uses the entropy condition for the system in an essential way 1995 john wiley sons inc

a meshless compositional approach to shape optimal design meshless computational strategies for shape optimal design through the composition of behavioral fields are developed in this paper the composition of behavioral fields is carried out in a manner that is procedurally similar to composition of primitive geometrical entities through set theoretic boolean operations in constructive solid geometry the methodology is shown to lead to a tighter integration between design and analysis and is termed constructive solid analysis a meshless approximation using non uniform rational b spline basis functions is used to discretize the behavioral fields defined over the geometrical primitives remeshing which is a critical bottleneck in shape optimization using conventional finite element analysis is either eliminated or limited to those primitives that are modified further it is demonstrated that the reconstruction of stiffness matrices is required only for those geometrical primitives that are affected by the design changes design sensitivity analysis techniques are developed for the discretized solution system arising out of the compositional analysis an adaptive numerical integration scheme is developed and shown to be necessary to accurately compute the effect of small domain perturbations on the problem functions some of the common challenges in conventional shape optimization problems such as difficulty in creating and merging multiple holes are naturally resolved using the developed compositional approach several representative problems are solved to validate and to demonstrate the methodology

multiscale phenomena green s functions the dirichlet to neumann formulation subgrid scale models bubbles and the origins of stabilized methods abstract an approach is developed for deriving variational methods capable of representing multiscale phenomena the ideas are first illustrated on the exterior problem for the helmholtz equation this leads to the well known dirichlet to neumann formulation next a class of subgrid scale models is developed and the relationships to bubble function methods and stabilized methods are established it is shown that both the latter methods are approximate subgrid scale models the identification for stabilized methods leads to an analytical formula for the intrinsic time scale whose origins have been a mystery heretofore

a construction procedure of iterative methods with cubical convergence we study the kantorovich convergence for parameter based methods for solving nonlinear operator equations in banach spaces we also derive a closed form of error bounds in terms of a real parameter a e 0 1

time stepping along characteristics with incomplete iteration for a galerkin approximation of miscible displacement in porous media miscible displacement of one incompressible fluid by another in a porous medium is modeled by a nonlinear coupled system of two partial differential equations the pressure equation is elliptic while the concentration equation is parabolic but normally convection dominated a sequential backward difference time stepping scheme is defined it approximates the pressure by a standard galerkin procedure and the concentration by a combination of a galerkin method and the method of characteristics optimal convergence rates in l 2 and h 1 are demonstrated for this scheme and for a modified version in which the algebraic equations at each time step are solved approximately by a limited number of preconditioned conjugate gradient iterations time stepping along the characteristics of the hyperbolic part of the concentration equation is shown to result in smaller time truncation errors than those of standard methods numerical results published elsewhere have confirmed that larger time steps are appropriate

on the transport diffusion algorithm and its applications to the navier stokes equations this paper deals with an algorithm for the solution of diffusion and or convection equations where we mixed the method of characteristics and the finite element method globally it looks like one does one step of transport plus one step of diffusion or projection but the mathematics show that it is also an implicit time discretization of thepde in lagrangian form we give an error bound h t h h t in the interesting case that holds also for the navier stokes equations even when the reynolds number is infinite euler equation

an optimal order error estimate for the discontinuous galerkin method in this paper a new approach is developed for analyzing the discontinuous galerkin method for hyperbolic equations for a model problem in r2 the method is shown to converge at a rate 0 hn l when applied with nth degree polynomial approxi mations over a semiuniform triangulation assuming sufficient regularity in the solution

a new algorithm for topology optimization using a level set method the level set method has been recently introduced in the field of shape optimization enabling a smooth representation of the boundaries on a fixed mesh and therefore leading to fast numerical algorithms however most of these algorithms use a hamilton jacobi equation to connect the evolution of the level set function with the deformation of the contours and consequently they can hardly create new holes in the domain at least in 2d in this work we propose an evolution equation for the level set function based on a generalization of the concept of topological gradient this results in a new algorithm allowing for all kinds of topology changes

numerical methods for convection dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures finite element and finite difference methods are combined with the method of characteristics to treat a parabolic problem of the form cu t bu x au x x f optimal order error estimates in l 2 and w 1 2 are derived for the finite element procedure various error estimates are presented for a variety of finite difference methods the estimates show that for convection dominated problems b gg a these schemes have much smaller time truncation errors than those of standard methods extensions to n space variables and time dependent or nonlinear coefficients are indicated along with applications of the concepts to certain problems described by systems of differential equations

a characteristic mixed method with dynamic finite element space for convection dominated diffusion problems abstract a method for numerically solving convection dominated diffusion problems using a combination of the method of characteristics with the mixed finite element method is presented and analyzed in this paper different numbers of elements and different basis functions are implemented at different time levels in order to accurately resolve time changing critical features this new method provides the ability to obtain approximations of the solution and its gradient simultaneously adopt large time steps and make grid refinements and basis function adjustments at any time necessary convergence analysis and error estimates are established furthermore comments are made on the efficiency and capability of the method for accurately solving transient problems with localized phenomena such as fronts shocks and boundary layers

the topological asymptotic for the helmholtz equation with dirichlet condition on the boundary of an arbitrarily shaped hole the aim of the topological sensitivity analysis is to obtain an asymptotic expansion of a design functional with respect to the creation of a small hole in the domain in this paper such an expansion is obtained for the helmholtz equation in two and three space dimensions with a dirichlet condition on the boundary of an arbitrarily shaped hole in this case the main difficulty is related to the nonhomogeneous symbol of the helmholtz operator in the numerical part of this work we will show that the topological sensitivity method is very promising for solving shape inverse problems in electromagnetic applications

a two grid eulerian lagrangian localized adjoint method to miscible displacement problems with dispersion term abstract in this paper we propose a novel two grid method of eulerian lagrangian localized adjoint method ellam mixed finite element method mfem solutions for nonlinear and coupled miscible displacement problems which describe complex fluid flow processes in porous media an ellam is used to solve the nonlinear convection diffusion equation for concentration and we present a mfem to compute the pressure equation for the pressure and darcy velocity a two grid scheme is investigated to linearize and decouple the mixed method equations it is illustrated that the coarse space can be selected as coarse as h o h 1 2 and the asymptotically optimal approximation as the nonlinear schemes can be still achieved as a result a large class of such nonlinear and coupled problems can be solved efficiently with large time steps

topological sensitivity analysis for three dimensional linear elasticity problem in this work we use the topological shape sensitivity method to obtain the topological derivative for three dimensional linear elasticity problems adopting the total potential energy as cost function and the equilibrium equation as constraint this method based on classical shape sensitivity analysis leads to a systematic procedure to calculate the topological derivative in particular firstly we present the mechanical model later we perform the shape derivative of the corresponding cost function and finally we calculate the final expression for the topological derivative using the topological shape sensitivity method and results from classical asymptotic analysis around spherical cavities in order to point out the applicability of the topological derivative in the context of topology optimization problems we use this information as a descent direction to solve a three dimensional topology design problem furthermore through this example we also show that the topological derivative together with an appropriate mesh refinement strategy are able to capture high quality shapes even using a very simple topology algorithm

a finite volume eulerian lagrangian localized adjoint method for solution of the advection dispersion equation a new mass conservative method for solution of the one dimensional advection dispersion equation is derived and discussed test results demonstrate that the finite volume eulerian lagrangian localized adjoint method fvellam outperforms standard finite difference methods in terms of accuracy and efficiency for solute transport problems that are dominated by advection for dispersion dominated problems the performance of the method is similar to that of standard methods like previous ellam formulations fvellam systematically conserves mass globally with all types of boundary conditions fvellam differs from other ellam approaches in that integrated finite differences instead of finite elements are used to approximate the governing equation this approach in conjunction with a forward tracking scheme greatly facilitates mass conservation the mass storage integral is numerically evaluated at the current time level and quadrature points are then tracked forward in time to the next level forward tracking permits straightforward treatment of inflow boundaries thus avoiding the inherent problem in backtracking as used by most characteristic methods of characteristic lines intersecting inflow boundaries fvellam extends previous ellam results by obtaining mass conservation locally on lagrangian space time elements details of the integration tracking and boundary algorithms are presented test results are given for problems in cartesian and radial coordinates

the corrected operator splitting approach applied to a nonlinear advection diffusion problem so called corrected operator splitting methods are applied to a 1 d scalar advection diffusion equation of buckley leverett type with general initial data front tracking and a 2nd order godunov method are used to advance the solution in time diffusion is modelled by piece wise linear finite elements at each new time level to obtain correct structure of shock fronts independently of the size of the time step a dynamically defined residual flux term is grouped with diffusion different test problems are considered and the methods are compared with respect to accuracy and runtime finally we extend the corrected operator splitting to 2 d equations by means of dimensional splitting and we apply it to a buckley leverett type problem including gravitational effects

an eulerian lagrangian solution technique for single phase compositional flow in three dimensional porous media we develop an eulerian lagrangian iterative solution technique for accurate and efficient numerical simulation of single phase compositional flow through three dimensional compressible porous media in the presence of multiple injection and production wells in the numerical simulator an eulerian lagrangian method is used to solve the convection diffusion transport equations and a mixed finite element method is used to solve the pressure equation numerical experiments are presented to investigate the performance of the method moreover because of the lagrangian nature of the algorithm the simulator is capable of using large time steps and coarse spatial grids to generate accurate and stable results

geometric constructions of iterative functions to solve nonlinear equations in this paper we present the geometrical interpretation of several iterative methods to solve a nonlinear scalar equation in addition we also review the extension to general banach spaces and some computational aspects of these methods

a level set method for topology optimization of heat conduction problem under multiple load cases in this paper we present a numerical approach of topology optimization under multiple load cases for heat conduction problem this framework is based on the theories of topological derivative and shape derivative for elliptic system we employ level set model to implicitly represent geometric boundary of thermal conductive material introducing topological derivative will generate new topology in the design domain which suppresses the dependence of initial topology guess to some extent the shape optimization is obtained by combining shape derivative with level set method the functional of quadratic temperature gradient is taken as the objective function in our analysis which is subjected to the state equation of steady heat conduction and volume constrain the shape of material domain is treated as the design variable and the final result is achieved by updating level set function gradually we develop an effective numerical technique to implement the optimal design with multiple load cases for heat conduction problem numerical examples demonstrate that our proposed approach is effective and robust for topology optimization of heat conduction problem

binary level set methods for topology and shape optimization of a two density inhomogeneous drum abstract we optimize eigenvalues in optimal shape design using binary level set methods the interfaces of subregions are represented implicitly by the discontinuities of binary level set functions taking two values 1 or 1 at convergence a binary constraint is added to the original model problems we propose two variational algorithms to solve the constrained optimization problems one is a hybrid type by coupling the lagrange multiplier approach for the geometry constraint with the augmented lagrangian method for the binary constraint the other is devised using the lagrange multiplier method for both constraints the two iterative algorithms are both largely independent of the initial guess and can satisfy the geometry constraint very accurately during the iterations intensive numerical results are presented and compared with those obtained by level set methods which demonstrate the effectiveness and robustness of our algorithms

a new method for inverse electromagnetic casting problems based on the topological derivative the inverse electromagnetic casting problem consists in looking for a suitable set of electric wires such that the electromagnetic field induced by an alternating current passing through them makes a given mass of liquid metal acquire a predefined shape in this paper we propose a new method for the topology design of such inductors the inverse electromagnetic casting problem is formulated as an optimization problem and topological derivatives are considered in order to locate new wires in the right position several numerical examples are presented showing that the proposed technique is effective to design suitable inductors

on the finite volume element method for general self adjoint elliptic problems the finite volume element method fve is a discretization technique for partial differential equations this paper develops discretization energy error estimates for general self adjoint elliptic boundary value problems with fve based on triangulations on which there exist linear finite element spaces and a very general type of control volumes covolumes the energy error estimates of this paper are also optimal but the restriction conditions for the covolumes given in r e bank and d j rose siam j numer anal 24 1987 pp 777 787 z q cai numer math 58 1991 pp 713 735 are removed the authors finally provide a counterexample to show that an expected l2 error estimate does not exist in the usual sense it is conjectured that the optimal order of u u h 0 omega should be o h for the general case

a eulerian lagrangian numerical scheme for the dispersion convection equation using conjugate space time grids abstract a new numerical scheme is proposed for the dispersion convection equation which combines the utility of a fixed grid in eulerian coordinates with the computational power of the lagrangian method convection is formally decoupled from dispersion in a manner which does not leave room for ambiguity the resulting convection problem is solved by the method of characteristics on a grid fixed in space dispersion is handled by finite elements on a separate grid which may but need not coincide wit the former at selected points in spacetime information is projected from one grid to another by local interpolation the conjugate grid method is implemented by linear finite elements in conjunction with piecewise linear interpolation functions and applied to five problems ranging from predominant dispersion to pure convection the results are entirely free of oscillations numerical dispersion exists but can be brought under control either by reducing the spatial increment or by increasing the time step size of the grid used to solve the convection problem contrary to many other methods best results are often obtained when the courant number exceeds 1

a probabilistic collocation eulerian lagrangian localized adjoint method on sparse grids for assessing co2 leakage through wells in randomly heterogeneous porous media abstract we develop a probabilistic collocation eulerian lagrangian localized adjoint method on sparse grids for assessing co2 leakage through wells in randomly heterogeneous porous media by utilizing the intrinsic mathematical numerical and physical properties of the mathematical model we model the process in which co2 is injected into a deep aquifer spreads within the aquifer and upon reaching a leaky well rises up to a shallower aquifer to quantify the leakage rate which depends on the pressure build up in the aquifer due to injection and the buoyancy of co2 the underlying eulerian lagrangian framework has high potential to improve the efficiency and accuracy for the numerical simulation of complex flow and transport processes in co2 sequestration the sparse grid probabilistic collocation framework adds computationally efficient uncertainty quantification functionality onto pre existing eulerian lagrangian methods in a nonintrusive manner it also provides a scalable framework to consider uncertainty in a straightforward parallel manner preliminary numerical experiments show the feasibility and potential of the method

classroom note geometry and convergence of euler s and halley s methods we investigate the global convergence of euler s and halley s methods by using their geometric interpretation a generalization of these methods is also briefly discussed

a stabilized mixed finite element method for darcy flow we develop new stabilized mixed finite element methods for darcy flow stability and an a priori error estimate in the stability norm are established a wide variety of convergent finite elements present themselves unlike the classical galerkin formulation which requires highly specialized elements an interesting feature of the formulation is that there are no mesh dependent parameters numerical tests confirm the theoretical results 2002 elsevier science b v all rights reserved

sensitivity of the macroscopic response of elastic microstructures to the insertion of inclusions this paper proposes an exact analytical formula for the topological sensitivity of the macroscopic response of elastic microstructures to the insertion of circular inclusions the macroscopic response is assumed to be predicted by a well established multi scale constitutive theory where the macroscopic strain and stress tensors are defined as volume averages of their microscopic counterpart fields over a representative volume element rve of material the proposed formula a symmetric fourth order tensor field over the rve domain is a topological derivative which measures how the macroscopic elasticity tensor changes when an infinitesimal circular elastic inclusion is introduced within the rve in the limits when the inclusion matrix phase contrast ratio tends to zero and infinity the sensitivities to the insertion of a hole and a rigid inclusion respectively are rigorously obtained the derivation relies on the topological asymptotic analysis of the predicted macroscopic elasticity and is presented in detail the derived fundamental formula is of interest to many areas of applied and computational mechanics to illustrate its potential applicability a simple finite element based example is presented where the topological derivative information is used to automatically generate a bi material microstructure to meet pre specified macroscopic properties

topological derivative for multi scale linear elasticity models applied to the synthesis of microstructures this paper proposes an algorithm for the synthesis optimization of microstructures based on an exact formula for the topological derivative of the macroscopic elasticity tensor and a level set domain representation the macroscopic elasticity tensor is estimated by a standard multi scale constitutive theory where the strain and stress tensors are volume averages of their microscopic counterparts over a representative volume element the algorithm is of simple computational implementation in particular it does not require artificial algorithmic parameters or strategies this is in sharp contrast with existing microstructural optimization procedures and follows as a natural consequence of the use of the topological derivative concept this concept provides the correct mathematical framework to treat topology changes such as those characterizing microstuctural optimization problems the effectiveness of the proposed methodology is illustrated in a set of finite element based numerical examples copyright 2010 john wiley sons ltd

some errors estimates for the box method we define and analyze several variants of the box method for discretizing elliptic boundary value problems in the plane our estimates show the error to be comparable to a standard galerkin finite element method using piecewise linear polynomials

optimal topological design through insertion and configuration of finite sized heterogeneities in this paper we develop a procedure for optimal topological design by sequentially inserting finite sized non spherical inclusions or holes within a homogeneous domain we propose a new criterion for topology change that results in a trade off problem to achieve the greatest least change in the objective for the least greatest change in the size of the inclusion hole respectively we derive the material derivative of the proposed objective termed as the configurational derivative that describes sensitivity of arbitrary functionals to arbitrary motions of the inclusion hole as well as the domain boundaries we specifically utilize the sensitivity to position orientation and scaling of finite sized heterogeneities to effect topological design we simplify the configurational derivative to the special case of infinitesimally small spherical inclusions or holes and show that the developed derivative is a generalization of the classical topological derivative the computational implementation relies on b spline isogeometric approximations we demonstrate through a series of examples optimal topology achieved through sequential insertion of a heterogeneity of fixed shape and optimization of its configuration location orientation and scale

solving the advection diffusion equation with the eulerian lagrangian localized adjoint method on unstructured meshes and non uniform time stepping eulerian lagrangian localized adjoint method ellam is used to solve the advection diffusion equation ade which is a very common mathematical model in physics in this work ellam is extended to triangular meshes standard integration schemes which perform well for rectangular grids are improved to reduce oscillations with unstructured triangulations numerical experiments for grid peclet numbers ranking from 1 to 100 show the efficiency of the developed scheme a new algorithm is also developed in order to avoid excessive numerical diffusion when using many time steps with the ellam the basic idea of this approach is to keep the same characteristics for all time steps and to interpolate only the concentration variations due to the dispersion process at the end of each time step although ellam requires a lot of integration points for unstructured meshes it remains a competitive method when using a single or many time steps compared to explicit discontinuous galerkin finite element method

a discontinuous finite volume method for a coupled fracture model abstract in this paper we present a discontinuous finite volume method for a hybrid dimensional coupled fracture model on triangular meshes in matrix and fracture the pressure is approximated by the discontinuous and continuous piecewise linear elements and in order to retain local conservation property the velocity is sought in the discontinuous lowest order raviart thomas element space and piecewise linear element space the proposed numerical scheme is only associated with the pressure so we avoid solving an indefinite saddle point problem in addition the discontinuous method eliminates the continuity of approximate functions across the interelement boundary so it has the advantage of high localizability and easy handling of complicated geometries optimal order error estimates for the pressure and velocity are proved and numerical experiments are carried out to confirm our theoretical analysis

the topological asymptotic expansion for the dirichlet problem the topological sensitivity analysis provides an asymptotic expansion of a shape function when creating a small hole inside a domain this expansion yields a descent direction which can be used for shape optimization if one wishes to keep a classical domain throughout the optimization process in this paper such an expansion is obtained for the poisson equation for a large class of cost functions and arbitrarily shaped holes in the three dimensional case this expansion depends on the shape of the hole but not on its orientation if the cost function involves only the solution u to the underlying partial differential equation whereas it may also depend on its orientation if the cost function involves the gradient nabla u in contrast the asymptotic expansion is independent of the shape in the two dimensional case a numerical example illustrates the use of the asymptotic expansion which yields a minimizing sequence of classical domains in a case where no classical solution exists

methods for reliable topology changes for perimeter regularized geometric inverse problems this paper is devoted to the incorporation of topological derivativelike expansions into level set methods for perimeter regularized geometric inverse problems the expansions are done up to the second order with respect to the lebesgue measure of the symmetric difference they provide simpler shape functionals still including the perimeter and therefore allow the construction of steepest descent and newton type algorithms to force topology changes during the level set evolution numerous numerical examples are provided that show the strong and also the weak points of the newly developed algorithms

sounding of finite solid bodies by way of topological derivative this paper is concerned with an application of the concept of topological derivative to elastic wave imaging of finite solid bodies containing cavities building on the approach originally proposed in the elastostatic theory of shape optimization the topological derivative which quantifies the sensitivity of a featured cost functional due to the creation of an infinitesimal hole in the cavity free reference body is used as a void indicator through an assembly of sampling points where it attains negative values the computation of topological derivative is shown to involve an elastodynamic solution to a set of supplementary boundary value problems for the reference body which are here formulated as boundary integral equations for a comprehensive treatment of the subject formulas for topological sensitivity are obtained using three alternative methodologies namely i direct differentiation approach ii adjoint field method and iii limiting form of the shape sensitivity analysis the competing techniques are further shown to lead to distinct computational procedures methodologies i and ii are implemented within a bem based platform and validated against an analytical solution a set of numerical results is included to illustrate the utility of topological derivative for 3d elastic wave sounding of solid bodies an approach that may perform best when used as a pre conditioning tool for more accurate gradient based imaging algorithms despite the fact that the formulation and results presented in this investigation are established on the basis of a boundary integral solution the proposed methodology is readily applicable to other computational platforms such as the finite element and finite difference techniques

on the topological derivative in shape optimization in this paper the topological derivative for an arbitrary shape functional is defined examples are provided for elliptic equations and the elasticity system in the plane the topological derivative can be used for solving shape optimization problems in structural mechanics

the shape and topological optimizations connection abstract in 1973 jean cea introduced a topological shape optimization algorithm based on a fixed point method the recent topological gradient concept provides a mathematical justification of cea s powerful method the aim of the topological gradient is to compute the sensitivity of a cost function when we drill a little hole in the domain some optimality conditions are derived when classical shape gradient and topological gradient are used simultaneously and numerical results are shown both in mechanics and electromagnetics

godunov mixed methods for advective flow problems in one space dimension a time splitting method for solving advection dominated parabolic partial differential equations is presented in this method a higher order godunov procedure approximates advection and a mixed finite element procedure approximates diffusion several variations on the basic scheme are formulated for solving one dimensional quasilinear parabolic problems with dirichlet boundary conditions a maximum principle for one variant of the scheme is demonstrated and l infty l 2 and l 2 l 2 error estimates for the approximate solution and the diffusive flux respectively are derived these estimates indicate that one variant of the scheme is l infty stable in certain situations but possibly suboptimal in error while another variant is optimal and l 2 stable

shape and topology optimization based on the phase field method and sensitivity analysis this paper discusses a structural optimization method that optimizes shape and topology based on the phase field method the proposed method has the same functional capabilities as a structural optimization method based on the level set method incorporating perimeter control functions the advantage of the method is the simplicity of computation since extra operations such as re initialization of functions are not required structural shapes are represented by the phase field function defined in the design domain and optimization of this function is performed by solving a time dependent reaction diffusion equation the artificial double well potential function used in the equation is derived from sensitivity analysis the proposed method is applied to two dimensional linear elastic and vibration optimization problems such as the minimum compliance problem a compliant mechanism design problem and the eigenfrequency maximization problem the numerical examples provided illustrate the convergence of the various objective functions and the effect that perimeter control has on the optimal configurations

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

a second order characteristic finite element scheme for convection diffusion problems a new characteristic finite element scheme is presented for it is of second order accuracy in time increment symmetric and unconditionally stable optimal error estimates are proved in the framework of l 2 theory numerical results are presented for two examples which show the advantage of the scheme