a nitsche stabilized finite element method for frictional sliding on embedded interfaces part ii intersecting interfaces abstract we extend the weighted nitsche s method proposed in the first part of this study to include multiple intersecting embedded interfaces these intersections arise either inside a computational domain where two internal interfaces intersect or on the boundary of the computational domain where an internal interface intersects with the external boundary we propose a variational treatment of both the interfacial kinematics and the external dirichlet constraints within nitsche s framework we modify the numerical analysis to account for these intersections and provide an explicit expression for the weights and the method parameters that arise in the nitsche variational form in the presence of junctions finally we demonstrate the performance of the method for both perfectly tied interfaces and perfectly plastic sliding interfaces through several benchmark examples

a reconstruction based chebyshev collocation method for the poisson equation an accurate treatment of the gibbs wilbraham phenomenon on irregular interfaces abstract an accurate numerical solution of the laplace equation and poisson equation is desirable in physical problems related to fluid flows and spectral methods are well suited towards this goal however in the presence of an interface the solution is discontinuous owing to the interface jump conditions and thus the numerical approximation is susceptible to the gibbs wilbraham phenomenon leading to a reduced solution accuracy in order to resolve the discontinuity at the interface we propose a reconstruction technique wherein the approximate solution is expressed as the sum of an infinitely differentiable smooth function and a modified heaviside function the smooth function is composed of chebyshev polynomials while the modified heaviside function is expressed in a weak form using the jump conditions across the interface the reconstruction framework allows us to impose the interfacial conditions exactly to motivate further application of the technique we first present the underlying rationale by revisiting the work of wilbraham 1848 13 followed by the numerical implementation we present numerical solution of four equations to illustrate the application of the proposed technique i an ordinary differential equation ode ii the euler bernoulli equation iii the laplace equation and iv the poisson equation the laplace and poisson equation are solved on a square domain embedded with complex interfacial geometries in two dimensions and on a cube with a spherical interface in three dimensions the method achieves error in the maximum norm o 10 14 using approximately 30 modes in each direction and requires a computational time of less than a second for all the problems we also conduct a sensitivity analysis to demonstrate the robustness of the algorithm under sub grid scale perturbations to interface location an alternate derivation for the modified heaviside function is proposed that is distinct from the taylor series expansion used to treat interface conditions in difference methods the pointwise error plots are subsequently presented to demonstrate that the gibbs wilbraham phenomenon is accurately resolved finally we present concluding thoughts on the potential applications of the reconstruction technique

the position code algorithm for contact searching a new global search method for general contact systems is developed and implemented in the dyna3d program along with a recent contact interface algorithm the concept of position codes for efficient global contact searching is presented with the position code algorithm the problem of sorting and searching in three dimensions is transformed to a process of sorting and searching within a one dimensional array the cost of contact searching is of the order of n log 2 n where n is the number of nodes in the system the proposed algorithms are uncomplicated and the implementation into any finite element code is straightforward

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

application of general orthogonal polynomials to the optimal control of time varying linear systems the operational matrices for forward or backward integration of general orthogonal polynomials are derived the tensor expression for the generalized orthogonal polynomial approximation of any two arbitrary functions is also introduced it is shown that the approximation solution obtained using the chebyshev polynomial is readily obtainable as a special case of the results derived thus the present results include results presented by shih in 1983 and chou and horng in 1984 a linear time varying optimal control system with a quadratic performance measure is solved by using the generalized orthogonal polynomials comparison of the results with those obtained using several different classical orthogonal polynomial approximations is also included

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

a contact domain method for large deformation frictional contact problems part 2 numerical aspects this second part of the work describes the numerical aspects of the developed contact domain method for large deformation frictional contact problems the theoretical basis of this contact method is detailed in the first part of this work starting from this the present contribution focuses on describing important algorithmic details that go along with the finite element implementation for two dimensional problems important aspects are the construction of the contact domain mesh via a constraint delaunay triangulation the linearization of the discretized contact contributions and some important technical aspects about the extrapolation procedure used for the predictive active set strategy finally a set of numerical examples is presented to demonstrate the performance of the developed contact strategy demanding static and dynamic contact problems in the context of large deformations including frictional effects as well as self contact show the wide applicability and the robustness of the proposed method

mixed finite elements for second order elliptic problems in three variables two families of mixed finite elements one based on simplices and the other on cubes are introduced as alternatives to the usual raviart thomas nedelec spaces these spaces are analogues of those introduced by brezzi douglas and marini in two space variables error estimates inl2 andh s are derived

a mortar segment to segment contact method for large deformation solid mechanics abstract contact modeling is still one of the most difficult aspects of non linear implicit structural analysis most 3d contact algorithms employed today use node on segment approaches for contacting dissimilar meshes two pass node on segment contact approaches have the well known deficiency of locking due to over constraint in this work we develop and demonstrate a segment to segment contact approach based on the mortar method by eliminating the over constraint this method appears to be much more robust than the node on segment approach furthermore node on segment approaches suffer when individual nodes slide out of contact at contact surface boundaries or when contacting nodes slide from facet to facet this causes jumps in the contact forces due to the discrete nature of the constraint enforcement and difficulties in convergence for implicit solution techniques these jumps in forces are avoided in the segment to segment approach since penetration is measured by the occluded weighted volume which varies smoothly with perturbations until now all mortar schemes have been developed for small deformation sliding and have mainly been applied on flat surfaces here we will present the integration scheme and the linearization approach used for handling the contact surface integrals in the event of large sliding the schemes are designed to conserve linear momentum and can be applied to arbitrarily curved 3d surfaces although expensive our numerical examples demonstrate that the approach is far more robust than even smoothed node on segment contact techniques that is problems previously not solvable with node on segment can now be solved with our segment to segment contact

a level set enhanced natural kernel contact algorithm for impact and penetration modeling summary a natural kernel contact nkc algorithm under the framework of the semi lagrangian reproducing kernel particle method semi lagrangian rkpm is proposed to model multi body contact with specific consideration for impact and penetration modeling the nkc algorithm utilizes the interaction of the semi lagrangian kernel functions associated with contacting bodies to serve as the non penetration condition the effects of friction are represented by introducing a layer of the friction like elasto plasticity material between contacting bodies this approach allows the frictional contact conditions and the associated kinematics to be naturally embedded in the semi lagrangian rkpm inter particle force calculation the equivalence in the karush kuhn tucker conditions between the proposed nkc algorithm and the conventional contact kinematic constraints as well as the associated state variable relationships are identified a level set method is further introduced in the nkc algorithm to represent the contact surfaces without pre defined potential contact surfaces the stability analysis performed in this work shows that temporal stability in the semi lagrangian rkpm with nkc algorithms is related to the velocity gradient between contacting bodies the proposed methods have been verified by several benchmark problems and applied to the simulation of impact and penetration processes copyright 2014 john wiley sons ltd

consistent treatment of boundaries with mortar contact formulations using dual lagrange multipliers abstract computational modelling of contact problems raises two basic questions which method should be used to enforce the contact conditions and how should this method be discretised the most popular enforcement methods are the lagrange multiplier method the penalty method and combinations of these two a frequently used discretisation method is the so called node to segment approach however this approach might lead to problems like jumps in contact forces loss of convergence or failure to pass the patch test thus in the last few years several segment to segment contact algorithms based on the mortar method were proposed combination of a mortar discretisation with a penalty based enforcement of the contact conditions leads to unphysical penetrations on the other hand a lagrange multiplier mortar method requires additional unknowns hence condensation of the lagrange multipliers is desirable to preserve the initial size of the system of equations this can be achieved by interpolating the lagrange multipliers with so called dual shape functions discretising two contacting bodies leads to opposed contact surface representations of finite element edges called slave and master elements respectively in current versions of dual lagrange multiplier mortar formulations an inconsistency at the boundary appears when only a part of a slave element instead of the entire element belongs to the contact area we present a modified definition of the dual shape functions in such slave elements the basic idea is to construct dual shape functions that fulfill the so called biorthogonality condition within the contact area this leads to consistent mortar matrices also in the boundary region to avoid ill conditioning of the stiffness matrix the modified mortar matrices are weighted with appropriate weighting factors in doing so the corresponding modified lagrange multiplier nodal values are of the same order as the unmodified ones various examples demonstrate the performance of the modified mortar contact algorithm

two dimensional mortar contact methods for large deformation frictional sliding this paper presents a mortar based formulation for the solution of two dimensional frictional contact problems involving finite deformation and large sliding as is widely recognized traditional node to surface contact formulations have several drawbacks in solution of deformable to deformable contact problems including lack of general patch test passage degradation of spatial convergence rates and robustness issues associated with the faceted representation of contacting surfaces the mortar finite element method initially proposed as a technique to join dissimilarly meshed domains has been shown to preserve optimal convergence rates in tied contact problems see discretization methods and iterative solvers based on domain decomposition springer verlag heidelberg 2001 for a recent review and is examined here as an alternative spatial discretization method for large sliding contact in particular a novel description for frictional sliding conditions in large deformation mortar formulations is proposed in this work in recent years the mortar element method has already been successfully implemented to solve frictional contact problems with linearized kinematics see int j numer meth engng 1993 36 3451 however in the presence of large deformations and finite sliding one must face difficulties associated with the definition and linearization of contact virtual work in the case where the mortar projection has a direct dependence on the tangential relative motion along the interface in this paper such a formulation is presented with particular emphasis on key aspects of the linearization procedure and on the robust description of the friction kinematics some novel techniques are proposed to treat the non smoothness in the contact geometry and the searching required to define mortar segments a number of numerical examples illustrate the performance and accuracy of the proposed formulation copyright 2004 john wiley sons ltd

application of shifted chebyshev series to the optimal control of linear distributed parameter systems by use of a particular property of the shifted chebyshev series the optimal control of a distributed parameter system is simplified into the optimal control of a linear time invariant lumped parameter system next a directly computational formulation for evaluating the optimal control and trajectory of a linear distributed parameter system is developed the formulation is straightforward and convenient for digital computation an illustrative example is given to demonstrate the applicability of the proposed method

analysis of linear distributed parameter systems via double fourier series double fourier series is used to approximate functions of two independent variables and then applied for the analysis of linear distributed parameter systems properties of fourier series are first briefly discussed and the operational matrix of integration and product are used to reduce the dynamical equation of a linear distributed system to a set of simultaneous linear algebraic equations the method is computationally very attractive and applications are demonstrated through illustrative examples

level set topology optimization of fluids in stokes flow we propose the level set method of topology optimization as a viable robust and efficient alternative to density based approaches in the setting of fluid flow the proposed algorithm maintains the discrete nature of the optimization problem throughout the optimization process leading to significant advantages over density based topology optimization algorithms specifically the no slip boundary condition is implemented directly this is accurate removes the need for interpolation schemes and continuation methods and gives significant computational savings by only requiring flow to be modeled in fluid regions topological sensitivity information is utilized to give a robust algorithm in two dimensions and familiar two dimensional power dissipation minimization problems are solved successfully computational efficiency of the algorithm is also clearly demonstrated on large scale three dimensional problems

resolution of fourth order problems by the mortar element method we present a resolution of fourth order problems by the mortar element method in the case of spectral discretizations we prove that this approach leads to a well posed discrete problem and the numerical analysis gives an optimal error estimate

approximation of the incompressible navier stokes equations using orthogonal subscale stabilization and pressure segregation on anisotropic finite element meshes abstract this paper describes a finite element model to solve the incompressible navier stokes equations based on the stabilization with orthogonal subscales and a pressure segregation the former consists of adding a least square form of the component orthogonal to the finite element space of the convective and pressure gradient terms this allows to deal with convection dominated flows and to use equal velocity pressure interpolation the pressure segregation is inspired in fractional step schemes although the converged solution corresponds to that of a monolithic time integration likewise we put special emphasis on the use of anisotropic grids in particular we describe some possible choices for the calculation of the element length that appears in the stabilization parameters and check their behavior in two classical numerical examples the preconditioning strategy used to solve the resulting algebraic system for very anisotropic meshes is also briefly described

two families of mixed finite elements for second order elliptic problems two families of mixed finite elements one based on triangles and the other on rectangles are introduced as alternatives to the usual raviart thomas nedelec spaces error estimates inl 2 andh 5 are derived for these elements a hybrid version of the mixed method is also considered and some superconvergence phenomena are discussed

analysis and identification of linear distributed systems via double general orthogonal polynomials abstract double general orthogonal polynomials are used for the analysis and parameter estimation of linear distributed systems the formulating of algorithms for analysing and for estimating the parameters of linear distributed systems is straightforward and convenient for digital computation illustrative examples are given to demonstrate the availability of the proposed method

a large deformation mortar formulation of self contact with finite sliding this paper presents a new numerical method in which self contact phenomena associated with a body undergoing large deformations and sliding can be described in particular the approach relies on a particular extension of the mortar approach appropriate for this class of problems a bounding volume hierarchy organized as a binary tree is built for the self contact surface based on the geometry and the mesh connectivity of the surface a curvature criterion using a new algorithm to detect subsurface adjacency is used to accelerate the self contact searching procedure to ensure that the mortar traction fields are properly defined on contiguous surface patches a novel facet sorting algorithm is also proposed based on the mesh connectivity of the contact element pairs found by the self contact searching algorithm several two and three dimensional numerical examples show the new self contact mortar formulation to be very efficient and also demonstrate that it can be combined with multi body contact algorithms to simulate a very general class of contact problems

the operational matrices of integration and differentiation for the fourier sine cosine and exponential series for the fourier sine cosine series basis vector varphi t and the fourier exponential series basis vector psi t a linear nonsingular transfor marion t is determined such that psi t t varphi t this result is then used to show that the operational matrices of integration p and q for varphi t and psi t respectively are related by the expression tp qt analogous results are derived for the corresponding operational matrices of differentiation d and r general expressions are derived for t p q d and r

a mortar based approach for the enforcement of periodic boundary conditions on arbitrarily generated meshes a new approach for the enforcement of periodic boundary conditions on representative volume element rve problems is presented the strategy is based on a mortar formulation and considers two and three dimensional problems in the finite deformation frame the mortar discretization is combined with the lagrange multiplier method for the enforcement of periodicity constraints at each pair of corresponding boundaries of the rve in a weak integral sense the interpolation of the lagrange multipliers is undertaken with dual shape functions which can be locally eliminated from the rve system of equations by static condensation avoiding an increase on the system size the numerical treatment within a newton based finite element solution procedure for the rve equilibrium problem is discussed in detail the resulting method is able to efficiently enforce periodic configurations over complex rves at finite strains with arbitrarily generated finite element meshes several numerical examples are presented to demonstrate the robustness of the method and the quality of results

a spectral element methodology tuned to parallel implementations abstract we present in this paper a modification of the conforming spectral element method for the approximation of the poisson equation this new procedure allows the implementation on a parallel machine in an easier and faster way the resulting method is nonconforming but the numerical analysis shows that the error between the exact solution and the discrete solution is of the same asymptotic order as the error of the original conforming method

applying nagata patches to smooth discretized surfaces used in 3d frictional contact problems the accurate solution of large deformation frictional contact problems using the finite element method is still a challenging task due to the strong nonlinearities involved this paper presents a smoothing method applicable to 3d contact surfaces discretized with an arbitrary mesh topology the quadratic nagata patch interpolation is adopted to define the smooth surface the resulting contact surface passes through all nodes of the mesh while providing a smooth description with at least g 1 continuity at the nodes and quasi g 1 continuity between the patches thus the proposed method avoids the non physical oscillations in the contact force which are induced by the traditionally used faceted contact surfaces description when slave nodes slide over several master segments moreover it allows the accurate evaluation of kinematic variables leading to important improvements in terms of convergence rate within the newton raphson iteration loop the developed global and local contact search algorithms designed for contact surfaces described by nagata patches are described in detail three numerical examples were selected to illustrate the advantages of the proposed smoothing method including a complex industrial example of sheet metal forming process the results show the significant improvements attained with the proposed approach in terms of efficiency robustness and accuracy when compared with the traditional faceted contact surfaces description 2014 published by elsevier b v

a stabilized mixed finite element method for darcy flow based on a multiscale decomposition of the solution recently masud and hughes proposed a stabilized mixed finite element formulation for darcy flow an interesting feature of this formulation is that there are no mesh dependent parameters in the present work we provide a derivation of this formulation based on a multiscale decomposition of the solution we also extend the work of masud and hughes to three dimensional problems and show the convergence rates for various three dimensional finite elements we also show that this formulation passes three dimensional constant flow patch tests for distorted element geometries i e elements with non constant jacobian robustness of this formulation is illustrated by performing numerical simulations on complex geometries

a discontinuous galerkin method with lagrange multipliers for the solution of helmholtz problems in the mid frequency regime we present a discontinuous galerkin method dgm for the solution of the helmholtz equation in the mid frequency regime our approach is based on the discontinuous enrichment method in which the standard polynomial field is enriched within each finite element by a non conforming field that contains free space solutions of the homogeneous partial differential equation to be solved hence for the helmholtz equation the enrichment field is chosen here as the superposition of plane waves we enforce a weak continuity of these plane waves across the element interfaces by suitable lagrange multipliers preliminary results obtained for two dimensional model problems discretized by uniform meshes reveal that the proposed dgm enables the development of elements that are far more competitive than both the standard linear and the standard quadratic galerkin elements for the discretization of helmholtz problems

virtual bubbles and galerkin least squares type methods ga l s abstract the equivalence between stabilized finite element methods or galerkin least squares tyoe methods ga l s and the standard galerkin method with bubble functions is established in an abstract framework the results are applicable to various finite element spaces including high order elements and applications to the advective diffusive model and to the stokes problem are presented illustrating the potential of the abstract theory introduced here

stabilized continuous and discontinuous galerkin techniques for darcy flow we design stabilized methods based on the variational multiscale decomposition of darcy s problem a model for the subscales is designed by using a heuristic fourier analysis this model involves a characteristic length scale that can go from the element size to the diameter of the domain leading to stabilized methods with different stability and convergence properties these sta bilized methods mimic the different possible functional settings of the continuous problem the optimal method depends on the velocity and pressure approximation order they also involve a subgrid projector that can be either the identity when applied to finite element residuals or can have an image orthogonal to the finite element space in pa rticular we have designed a new stabilized method that allows the use of piecewise constant pressures we consider a general setting in which velocity and pressure can be approximated by either continuous or discontinuous approximations all these methods have been analyzed proving stability and convergence results in some cases duality arguments have been used to obtain error bounds in the l 2 norm

a new stabilized finite element formulation for scalar convection diffusion problems the streamline and approximate upwind petrov galerkin method abstract a new stabilized and accurate finite element formulation for convection dominated problems is herein developed the basis of the new formulation is the choice of a new upwind function the upwind function chosen for the new method provokes its degeneration into the supg or cau methods depending on the approximate solution s regularity the accuracy and stability of the new formulation for the linear and scalar advection diffusion equation is demonstrated in several numerical examples

unified stabilized finite element formulations for the stokes and the darcy problems in this paper we propose stabilized finite element methods for both stokes and darcy s problems that accommodate any interpolation of velocities and pressures apart from the interest of this fact the important issue is that we are able to deal with both problems at the same time in a completely unified manner in spite of the fact that the functional setting is different concerning the stabilization formulation we discuss the effect of the choice of the length scale appearing in the expression of the stabilization parameters both in what refers to stability and to accuracy this choice is shown to be crucial in the case of darcy s problem as an additional feature of this work we treat two types of stabilized formulations showing that they have a very similar behavior

a stabilized mixed discontinuous galerkin method for darcy flow abstract a new mixed stabilized discontinuous galerkin formulation for darcy flow is presented the formulation combines several attributes not simultaneously satisfied by other methods it is convergent for any combination of velocity and pressure interpolation higher than first order it exactly satisfies a mass balance on each element and it passes two and three dimensional constant flow patch tests for distorted element geometries the key ingredient in the formulation is a volumetric residual based stabilization term that does not involve any mesh dependent parameters

the non symmetric nitsche method for the parameter free imposition of weak boundary and coupling conditions in immersed finite elements abstract we explore the use of the non symmetric nitsche method for the weak imposition of boundary and coupling conditions along interfaces that intersect through a finite element mesh in contrast to symmetric nitsche methods it does not require stabilization and therefore does not depend on the appropriate estimation of stabilization parameters we first review the available mathematical background recollecting relevant aspects of the method from a numerical analysis viewpoint we then compare accuracy and convergence of symmetric and non symmetric nitsche methods for a laplace problem a kirchhoff plate and in 3d elasticity our numerical experiments confirm that the non symmetric method leads to reduced accuracy in the l 2 error but exhibits superior accuracy and robustness for derivative quantities such as diffusive flux bending moments or stress based on our numerical evidence the non symmetric nitsche method is a viable alternative for problems with diffusion type operators in particular when the accuracy of derivative quantities is of primary interest

modeling flow in porous media with double porosity permeability a stabilized mixed formulation error analysis and numerical solutions abstract the flow of incompressible fluids through porous media plays a crucial role in many technological applications such as enhanced oil recovery and geological carbon dioxide sequestration the flow within numerous natural and synthetic porous materials that contain multiple scales of pores cannot be adequately described by the classical darcy equations it is for this reason that mathematical models for fluid flow in media with multiple scales of pores have been proposed in the literature however these models are analytically intractable for realistic problems in this paper a stabilized mixed four field finite element formulation is presented to study the flow of an incompressible fluid in porous media exhibiting double porosity permeability the stabilization terms and the stabilization parameters are derived in a mathematically consistent manner and the computationally convenient equal order interpolation of all the field variables is shown to be stable a systematic error analysis is performed on the resulting stabilized weak formulation representative problems patch tests and numerical convergence analyses are performed to illustrate the performance and convergence behavior of the proposed mixed formulation in the discrete setting the accuracy of numerical solutions is assessed using the mathematical properties satisfied by the solutions of this double porosity permeability model moreover it is shown that the proposed framework can perform well under transient conditions and that it can capture well known instabilities such as viscous fingering

imposing dirichlet boundary conditions with nitsche s method and spline based finite elements a key challenge while employing non interpolatory basis functions in finite element methods is the robust imposition of dirichlet boundary conditions the current work studies the weak enforcement of such conditions for b spline basis functions with application to both second and fourth order problems this is achieved using concepts borrowed from nitsche s method which is a stabilized method for imposing constraints on surfaces conditions for the stability of the system of equations are derived for each class of problem stability parameters in the nitsche weak form are then evaluated by solving a local generalized eigenvalue problem at the dirichlet boundary the approach is designed to work equally well when the grid used to build the splines conforms to the physical boundary of interest as well as to the more general case when it does not through several numerical examples the approach is shown to yield optimal rates of convergence copyright 2010 john wiley sons ltd

node to node scheme for three dimensional contact mechanics using polyhedral type variable node elements abstract a node to node contact scheme which is applicable for three dimensional contact analyses involving large deformations is developed with the aid of polyhedral elements the key issue is to transform nonmatching meshes into matching meshes in a seamless manner because polyhedral elements are allowed to have arbitrary numbers of polygonal faces and nodes they can be used as transition elements for coupling nonmatching meshes in this paper the polyhedral elements make it possible to always maintain node to node contact during the contact deformation the present approach guarantees that the patch test is passed and the nonpenetration condition is satisfied and hence it yields smoother contact pressure with faster convergence than the conventional node to surface or surface to surface contact scheme

a constrained optimization methodology for the detection phase in contact mechanics simulations the detection phase in computational contact mechanics can be subdivided into a global search and a local detection when potential contact is detected by the former a rigorous local detection determines which surface elements come or may come in contact in the current increment we first introduce a rigorous definition of the closest point for non differentiable lower dimensional manifolds we then simplify the detection by formulating an optimization problem subject to inequality constraints the formulation is then solved using different techniques from the field of mathematical optimization for both linear and quadratic finite element meshes the resulting general and robust detection scheme is tested on a set of problems and compared with other techniques commonly used in computational geometry

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