coupling an eulerian fluid calculation to a lagrangian solid calculation with the ghost fluid method we propose a numerical method for modeling multimaterial flows where the domain is decomposed into separate eulerian and lagrangian subdomains that is the equations are written in eulerian form in one subdomain and in lagrangian form in the other subdomain this is of interest for example when considering the effect of underwater explosions on the hull of a ship or the impact of a low speed projectile on a soft explosive target on the one hand high speed fluid flows are traditionally modeled by applying shock capturing schemes to the compressible euler equations to avoid problems associated with tangling of a lagrangian mesh on the other hand solid dynamics calculations are traditionally carried out using lagrangian numerical methods to avoid problems associated with numerical smearing in eulerian calculations we use the ghost fluid method to create accurate discretizations across the eulerian lagrangian interface the numerical method is presented in both one and two spatial dimensions three dimensional extensions to the interface coupling method are straightforward

computability structure of the sobolev spaces and its applications in this paper we present a definition of s computability for functions in sobolev spaces hs rd s r two applications of s computability are then discussed one is to address several open problems regarding classical computable functions the other is to conduct computable analysis of several important classes of partial differential equations

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

numerical study of the turbulent flow past an airfoil with trailing edge separation presentation d une methode numerique par volume fini pour la resolution des equations de navier stokes bidimensionnelles incompressibles et stationnaires en coordonnees generales curvilignes application de la methode aux ecoulements turbulents sur des profils avec et sans separation au bord de sortie posterieur comparaison des calculs avec des donnees experimentales

computational study of subcritical response in flow past a circular cylinder flow past a circular cylinder is investigated in the subcritical regime below the onset of benard von karman vortex shedding at reynolds number re c similar or equal to 47 the transient response of infinitesimal perturbations is computed the domain requirements for obtaining converged results is discussed at length it is shown that energy amplification occurs as low as re 2 2 throughout much of the subcritical regime the maximum energy amplification increases approximately exponentially in the square of re reaching 6800 at re c the spatiotemporal structure of the optimal transient dynamics is shown to be transitory benard von karman vortex streets at re similar or equal to 42 the long time structure switches from exponentially increasing downstream to exponentially decaying downstream three dimensional computations show that two dimensional structures dominate the energy growth except at short times

the computational complexity of maximization and integration on montre que ces deux problemes de la theorie de la complexite de calcul aussi bien que les autres resolvant la maximisation et l integration sont equivalents aux problemes de complexite de calcul discret naturel dont la plupart ont deja beaucoup ete etudies

a particle level set based sharp interface cartesian grid method for impact penetration and void collapse an eulerian sharp interface cartesian grid method is developed for the numerical simulation of the response of materials to impact shocks and detonations the mass momentum and energy equations are solved along with evolution equations for deviatoric stresses and equivalent plastic strain these equations are cast in eulerian conservation law form the mie gruneisen equation of state is used to obtain pressure and the material is modeled as a johnson cook solid the eno scheme is employed to capture shocks in combination with a hybrid particle level set technique to evolve sharp immersed boundaries the numerical technique is able to handle collisions between multiple materials and can accurately compute the dynamics of the immersed boundaries results of calculations for axisymmetric taylor bar impact and penetration of a tungsten rod into steel plate show good agreement with moving finite element solutions and experimental results qualitative agreement with theory is shown for the void collapse phenomenon in an impacted material containing a spherical void

fictitious domain approach with hp finite element approximation for incompressible fluid flow we consider the application of fictitious domain approach combined with least squares spectral elements for the numerical solution of fluid dynamic incompressible equations fictitious domain methods allow problems formulated on a complicated shaped domain to be solved on a simpler domain containing least squares spectral element method has been used to develop the discrete model as this scheme combines the generality of finite element methods with the accuracy of spectral methods moreover the least squares methods have theoretical and computational advantages in the algorithmic design and implementation this paper presents the formulation and validation of the fictitious domain least squares spectral element approach for the steady incompressible navier stokes equations the convergence of the approximated solution is verified solving two dimensional benchmark problems demonstrating the predictive capability of the proposed formulation

a cut cell non conforming cartesian mesh method for compressible and incompressible flow this paper details a multigrid accelerated cut cell non conforming cartesian mesh methodology for the modelling of inviscid compressible and incompressible flow this is done via a single equation set that describes sub trans and supersonic flows cut cell technology is developed to furnish body fitted meshes with an overlapping mesh as starting point and in a manner which is insensitive to surface definition inconsistencies spatial discretization is effected via an edge based vertex centred finite volume method an alternative dual mesh construction strategy similar to the cell centred method is developed incompressibility is dealt with via an artificial compressibility algorithm and stabilization achieved with artificial dissipation in compressible flow shocks are captured via pressure switch activated upwinding the solution process is accelerated with full approximation storage fas multigrid where coarse meshes are generated automatically via a volume agglomeration methodology this is the first time that the proposed discretization and solution methods are employed to solve a single compressible incompressible equation set on cut cell cartesian meshes the developed technology is validated by numerical experiments the standard discretization and alternative methods were found equivalent in accuracy and computational cost the multigrid implementation achieved decreases in cpu time of up to one order of magnitude copyright 2007 john wiley sons ltd

treatment of contact separation in eulerian high speed multimaterial dynamic simulations summary a frictionless contact separation treatment in a sharp interface eulerian framework is presented to handle the general situation of high speed impact and separation of materials the algorithm has been developed for an established eulerian based cartesian grid multimaterial flow code in which the interfaces are tracked in a sharp manner using a standard narrow band level set approach boundary conditions have been applied using a modified ghost fluid method for elasto plastic materials the sharp interface treatment maintains the distinct interacting interfaces without smearing the contact zone while also removing the difficulties associated with lagrangian moving mesh entities in contact separation situations the algorithm has been tested and verified against experimental and numerical results for three different problems in the high strain rate regime which involve contact separation and sliding of materials copyright 2014 john wiley sons ltd

capturing shock reflections godunov type schemes based on exact or approximate solutions to the riemann problem have proven to be an excellent tool to compute approximate solutions to hyperbolic systems of conservation laws however there are many instances in which a particular scheme produces inappropriate results in this paper we consider several situations in which roe s scheme gives incorrect results or blows up all together and we propose an alternative flux formula that produces numerical approximations in which the pathological behavior is either eliminated or reduced to computationally acceptable levels

jordan curves with polynomial inverse moduli of continuity computational complexity of two dimensional domains whose boundaries are polynomial time computable jordan curves with polynomial inverse moduli of continuity is studied it is shown that the membership problem of such a domain can be solved in p n p i e in polynomial time relative to an oracle in np in contrast to the higher upper bound p m p for domains without the property of polynomial inverse modulus of continuity on the other hand the lower bound of up for the membership problem still holds for domains with polynomial inverse moduli of continuity it is also shown that the shortest path problem of such a domain can be solved in pspace close to its known lower bound while no fixed upper bound was known for domains without this property

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

isogeometric residual minimization igrm for non stationary stokes and navier stokes problems abstract we show that it is possible to obtain a linear computational cost fem based solver for non stationary stokes and navier stokes equations our method employs a technique developed by guermond and minev 2011 which consists of singular perturbation plus a splitting scheme while the time integration schemes are implicit we use finite elements to discretize the spatial counterparts at each time step we solve a pde having weak derivatives in one direction only which allows for the linear computational cost at the expense of handling strong second order derivatives of the previous time step solution on the right hand side of these pdes this motivates the use of smooth functions such as b splines for high reynolds numbers some of these pdes become unstable to deal robustly with these instabilities we propose to use a residual minimization technique we test our method on problems having manufactured solutions as well as on the cavity flow problem

fast isogeometric solvers for hyperbolic wave propagation problems abstract we use the alternating direction method to simulate implicit dynamics our spatial discretization uses isogeometric analysis namely we simulate a hyperbolic wave propagation problem in which we use tensor product b splines in space and an implicit time marching method to fully discretize the problem we approximate our discrete operator as a kronecker product of one dimensional mass and stiffness matrices as a result of this algebraic transformation we can factorize the resulting system of equations in linear i e o n time at each step of the implicit method we demonstrate the performance of our method in the model p wave propagation problem we then extend it to simulate the linear elasticity problem once we decouple the vector problem using alternating triangular methods we prove theoretically and experimentally the unconditional stability of both methods

a conservative immersed interface method for large eddy simulation of incompressible flows we propose a conservative second order accurate immersed interface method for representing incompressible fluid flows over complex three dimensional solid obstacles on a staggered cartesian grid the method is based on a finite volume discretization of the incompressible navier stokes equations which is modified locally in cells that are cut by the interface in such a way that accuracy and conservativity are maintained a level set technique is used for description and tracking of the interface geometry so that an extension of the method to moving boundaries and flexible walls is straightforward numerical stability is ensured for small cells by a conservative mixing procedure discrete conservation and sharp representation of the fluid solid interface render the method particularly suitable for large eddy simulations of high reynolds number flows accuracy second order grid convergence and robustness of the method is demonstrated for several test cases inclined channel flow at re 20 flow over a square cylinder at re 100 flow over a circular cylinder at re 40 re 100 and re 3900 as well as turbulent channel flow with periodic constrictions at re 10 595

sharp interface cartesian grid method ii a technique for simulating droplet interactions with surfaces of arbitrary shape a fixed grid sharp interface method is developed to simulate droplet impact and spreading on surfaces of arbitrary shape a finite difference technique is used to discretize the incompressible navier stokes equations on a cartesian grid to compute flow around embedded solid boundaries a previously developed sharp interface method for solid immersed boundaries is used the ghost fluid method gfm is used for fluid fluid interfaces the model accounts for the effects of discontinuities such as density and viscosity jumps and singular sources such as surface tension in both bubble and droplet simulations with a level set representation of the propagating interface large deformations of the boundary can be handled easily the model successfully captures the essential features of interactions between fluid fluid and solid fluid phases during impact and spreading moving contact lines are modeled with contact angle hysteresis and contact line motion on non planar surfaces is computed experimental observations and other simulation results are used to validate the calculations

isogeometric analysis of the isothermal navier stokes korteweg equations this paper is devoted to the numerical simulation of the navier stokes korteweg equations a phase eld model for water water vapor two phase ows we develop a numerical formulation based on isogeometric analysis that permits straightforward treatment of the higher order partial dierenti al operator that represents capillarity we introduce a new renement methodology that desensitizes the numerical solution to the computational mesh and achieves mesh invariant solutions finally we present several numerical examples in two and three dimensions that illustrate the eectiveness and robustness of our approach

an isogeometric analysis approach to gradient damage models continuum damage formulations are commonly used for the simulation of diffuse fracture processes implicit gradient damage models are employed to avoid the spurious mesh dependencies associated with local continuum damage models the c 0 continuity of traditional finite elements has hindered the study of higher order gradient damage approximations in this contribution we use isogeometric finite elements which allow for the construction of higher order continuous basis functions on complex domains we study the suitability of isogeometric finite elements for the discretization of higher order gradient damage approximations

direct numerical simulation of flow past elliptic cylinders flow over elliptic cylinders can be considered prototypical of flow over a range of bluff bodies since the geometry allows one to study the effect of both thickness and angle of attack on the flow field therefore a careful study of this flow should provide valuable insight into the phenomenon of unsteady separation and the structure of bluff body wakes with this in mind a spectral collocation technique has been developed to simulate the full three dimensional incompressible flow over elliptic cylinders and unlike spectral element and spectral multidomain techniques here the flow is solved in a single domain the equations are discretized on a body fitted elliptic cylindrical grid and properties of the metric associated with this coordinate system are used to solve the governing equations in an efficient manner key issues including the inflow and outflow boundary conditions and time discretization are discussed in detail with the hope that this will facilitate future simulations of similar flows finally we present results of two and three dimensional simulations for a range of flow and geometric parameters the results are compared with available experimental data and it is found that important quantities like strouhal numbers and drag coefficients match well with established values

a generalized finite element formulation for arbitrary basis functions from isogeometric analysis to xfem many of the formulations of cm rent research interest including iosogeometric methods and the extended finite element method use nontraditional basis functions some such as subdivision surfaces may not have convenient analytical representations the concept of an element if appropriate at all no longer coincides with the traditional definition developing a new software for each new class of basis functions is a large research burden especially if the problems involve large deformations non linear materials and contact the objective of this paper is to present a method that separates as much as possible the generation and evaluation of the basis functions from the analysis resulting in a formulation that can be implemented within the traditional structure of a finite clement program but that permits the use of arbitrary sets of basis functions that are defined only through the input file elements ranging from a traditional linear four node tetrahedron through a higher order element combining xfem and isogeometric analysis may be specified entirely through an input file without any additional programming examples of this framework to applications with lagrange elements isogeometric elements and xfem basis functions for fracture are presented

the ls stag method a new immersed boundary level set method for the computation of incompressible viscous flows in complex moving geometries with good conservation properties this paper concerns the development of a new cartesian grid immersed boundary ib method for the computation of incompressible viscous flows in two dimensional irregular geometries in ib methods the computational grid is not aligned with the irregular boundary and of upmost importance for accuracy and stability is the discretization in cells which are cut by the boundary the so called cut cells in this paper we present a new ib method called the ls stag method which is based on the mac method for staggered cartesian grids and where the irregular boundary is sharply represented by its level set function this implicit representation of the immersed boundary enables us to calculate efficiently the geometry parameters of the cut cells we have achieved a novel discretization of the fluxes in the cut cells by enforcing the strict conservation of total mass momentum and kinetic energy at the discrete level our discretization in the cut cells is consistent with the mac discretization used in cartesian fluid cells and has the ability to preserve the five point cartesian structure of the stencil resulting in a highly computationally efficient method the accuracy and robustness of our method is assessed on canonical flows at low to moderate reynolds number taylor couette flow flows past a circular cylinder including the case where the cylinder has forced oscillatory rotations finally we will extend the ls stag method to the handling of moving immersed boundaries and present some results for the transversely oscillating cylinder flow in a free stream

the discrete continuity equation in primitive variable solutions of incompressible flow abstract the use of a non staggered computational grid for the numerical solutions of the incompressible flow equations has many advantages over the use of a staggered grid a penalty however is inherent in the finite difference approximations of the governing equations on non staggered grids in the primitive variable solutions the penalty is that the discrete continuity equation does not converge to machine accuracy rather it converges to a source term which is proportional to the fourth order derivative of the pressure the time increment and the square of the grid spacing an approach which minimizes the error in the discrete continuity equation is developed numerical results obtained for the driven cavity problem confirm the analytical developments

computation of solid liquid phase fronts in the sharp interface limit on fixed grids a finite difference formulation is applied to track solid liquid boundaries on a fixed underlying grid the interface is not of finite thickness but is treated as a discontinuity and is explicitly tracked the imposition of boundary conditions exactly on a sharp interface that passes through the cartesian grid is performed using simple stencil readjustments in the vicinity of the interface attention is paid to formulating difference schemes that are globally second order accurate in x and t error analysis and grid refinement studies are performed for test problems involving the diffusion and convection diffusion equations and for stable solidification problems issues concerned with stability and change of phase of grid points in the evolution of solid liquid phase fronts are also addressed it is demonstrated that the field calculation is second order accurate while the position of the phase front is calculated to first order accuracy furthermore the accuracy estimates hold for the cases where there is a property jump across the interface unstable solidification phenomena are simulated and an attempt is made to compare results with previously published work the results indicate the need to begin an effort to benchmark computations of instability phenomena

evaluation of multifidelity surrogate modeling techniques to construct closure laws for drag in shock particle interactions abstract meta or surrogate models constructed from meso scale simulations can be used in place of empirical correlations to close macro scale equations in shocked particulate flows surrogate models for drag are constructed as functions of shock mach number ma particle volume fraction reynolds number re etc the computational cost of the high fidelity meso scale simulations is a challenge in construction of surrogates in such hierarchical multi scale frameworks here multifidelity surrogate modeling techniques are evaluated as inexpensive alternatives to high fidelity surrogate models for obtaining closure laws for drag in shock particle interactions preliminary surrogates for drag as a function of ma and are constructed from ensembles of low fidelity coarse grid mesoscale computations the low fidelity surrogates are subsequently corrected using only a few n h f high fidelity computations to obtain multifidelity surrogate models the paper evaluates three different methods for correcting an initial low fidelity surrogate space mapping sm radial basis functions rbf and modified bayesian kriging mbkg of these methods mbkg is found to provide the best multi fidelity surrogate model simultaneously minimizing the computational cost and error in the constructed surrogate

a cartesian grid method for modeling multiple moving objects in 2d incompressible viscous flow we present an efficient method for solving 2d incompressible viscous flows around multiple moving objects our method employs an underlying regular cartesian grid to solve the system using a streamfunction vorticity formulation and with discontinuities representing the embedded objects the no penentration condition for the moving geometry is satisfied by superposing a homogenous solution to the poisson s equation for the streamfunction the no slip condition is satisfied by generating vorticity on the surfaces of the objects both the initial poisson solution and the evaluation of the homogenous solution require embedding irregular discontinuities in a fast poisson solver computation time is dictated by the time required to do a fast poisson solution plus solve an integral form of laplace s equation there is no significant increase in computational cost if the geometry of the embedded objects is variable and moving relative to the underlying grid we test the method against the canonical example of flow past a cylinder and obtained new results on the flow and forces of two cylinders moving relative to each other

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

computational cost estimates for parallel shared memory isogeometric multi frontal solvers abstract in this paper we present computational cost estimates for parallel shared memory isogeometric multi frontal solvers the estimates show that the ideal isogeometric shared memory parallel direct solver scales as o p 2 log n p for one dimensional problems o n p 2 for two dimensional problems and o n 4 3 p 2 for three dimensional problems where n is the number of degrees of freedom and p is the polynomial order of approximation the computational costs of the shared memory parallel isogeometric direct solver are compared with those corresponding to the sequential isogeometric direct solver being the latest equal to o n p 2 for the one dimensional case o n 1 5 p 3 for the two dimensional case and o n 2 p 3 for the three dimensional case the shared memory version significantly reduces both the scalability in terms of n and p theoretical estimates are compared with numerical experiments performed with linear quadratic cubic quartic and quintic b splines in one and two spatial dimensions

alternate formulations for the pressure equation laplacian on a collocated grid for solving the unsteady incompressible navier stokes equations abstract the solution of the unsteady incompressible navier stokes on a collocated grid is investigated a consistent formulation to construct the laplacian in the pressure equation is developed it is shown that within this framework laplacians with different characteristics can be constructed which can be made to satisfy discrete continuity and considerably reduce pressure oscillations a third order forward biased approximation for the gradient operator of pressure works very well in completely eliminating pressure oscillations in a lid driven cavity flow test problem

a differentially interpolated direct forcing immersed boundary method for predicting incompressible navier stokes equations in time varying complex geometries a dispersion relation preserving dual compact scheme developed in cartesian grids is applied together with the immersed boundary method to solve the flow equations in irregular and time varying domains the artificial momentum forcing term applied at certain points in cells containing fluid and solid allows an imposition of velocity condition to account for the motion of solid body we develop in this study a differential based interpolation scheme which can be easily extended to three dimensional simulation the results simulated from the proposed immersed boundary method agree well with other numerical and experimental results for the chosen benchmark problems the accuracy and fidelity of the ib flow solver developed to predict flows with irregular boundaries are therefore demonstrated

ghost fluid method for strong shock interactions part 2 immersed solid boundaries numerical simulation of shock waves interacting with multimaterial interface is immensely challenging particularly when the embedded interface is retained as a sharp entity the challenge lies in accurately capturing and representing the interface dynamics and the wave patterns at the interface in this regard the ghost fluid method has been successfully used to capture the interface conditions for both fluid fluid and solid fluid interfaces however the ghost fluid method results in over underheating errors when shocks impact interfaces and hence must be supplemented with numerical corrective measures to mitigate these errors such corrections typically fail for strong shock applications variants and extensions ofthe ghost fluid method have been proposed to remedy its shortcomings with mixed success in this paper the performance of approaches based on the ghost fluid method in the case of strong shocks impinging on immersed solid boundaries in compressible flows is evaluated it is found that from the viewpoint of simplicity robustness and accuracy a reflective boundary condition used in conjunction with a local riemann solver at the interface proves to be a good choice the method is found to be stable accurate and robust for wide range problems involving strong shocks interacting with embedded solid objects

isogeometric analysis of the cahn hilliard phase field model abstract the cahn hilliard equation involves fourth order spatial derivatives finite element solutions are not common because primal variational formulations of fourth order operators are only well defined and integrable if the finite element basis functions are piecewise smooth and globally c 1 continuous there are a very limited number of two dimensional finite elements possessing c 1 continuity applicable to complex geometries but none in three dimensions we propose isogeometric analysis as a technology that possesses a unique combination of attributes for complex problems involving higher order differential operators namely higher order accuracy robustness two and three dimensional geometric flexibility compact support and most importantly the possibility of c 1 and higher order continuity a nurbs based variational formulation for the cahn hilliard equation was tested on two and three dimensional problems we present steady state solutions in two dimensions and for the first time in three dimensions to achieve these results an adaptive time stepping method is introduced we also present a technique for desensitizing calculations to dependence on mesh refinement this enables the calculation of topologically correct solutions on coarse meshes opening the way to practical engineering applications of phase field methodology

an immersed interface method for viscous incompressible flows involving rigid and flexible boundaries we present an immersed interface method for the incompressible navier stokes equations capable of handling both rigid and flexible boundaries the immersed boundaries are represented by a number of lagrangian control points in order to ensure that the no slip condition on the rigid boundary is satisfied singular forces are applied on the fluid the forces are related to the jumps in pressure and the jumps in the derivatives of both pressure and velocity and are interpolated using cubic splines the strength of the singular forces at the rigid boundary is determined by solving a small system of equations at each timestep for flexible boundaries the forces that the boundary exerts on the fluid are computed from the constitutive relation of the flexible boundary and are applied to the fluid through the jump conditions the position of the flexible boundary is updated implicitly using a quasi newton method bfgs within each timestep the navier stokes equations are discretized on a staggered cartesian grid by a second order accurate projection method for pressure and velocity and the overall scheme is second order accurate

a finite strain eulerian formulation for compressible and nearly incompressible hyperelasticity using high order b spline finite elements summary we present a numerical formulation aimed at modeling the nonlinear response of elastic materials using large deformation continuum mechanics in three dimensions this finite element formulation is based on the eulerian description of motion and the transport of the deformation gradient when modeling a nearly incompressible solid the transport of the deformation gradient is decomposed into its isochoric part and the jacobian determinant as independent fields a homogeneous isotropic hyperelastic solid is assumed and b splines based finite elements are used for the spatial discretization a variational multiscale residual based approach is employed to stabilize the transport equations the performance of the scheme is explored for both compressible and nearly incompressible applications the numerical results are in good agreement with theory illustrating the viability of the computational scheme copyright 2011 john wiley sons ltd

a level set approach to eulerian lagrangian coupling we present a numerical method for coupling an eulerian compressible flow solver with a lagrangian solver for fast transient problems involving fluid solid interactions such coupling needs arise when either specific solution methods or accuracy considerations necessitate that different and disjoint subdomains be treated with different eulerian or lagrangian schemes the algorithm we propose employs standard integration of the eulerian solution over a cartesian mesh to treat the irregular boundary cells that are generated by an arbitrary boundary on a structured grid the eulerian computational domain is augmented by a thin layer of cartesian ghost cells boundary conditions at these cells are established by enforcing conservation of mass and continuity of the stress tensor in the direction normal to the boundary the description and the kinematic constraints of the eulerian boundary rely on the unstructured lagrangian mesh the lagrangian mesh evolves concurrently driven by the traction boundary conditions imposed by the eulerian counterpart several numerical tests designed to measure the rate of convergence and accuracy of the coupling algorithm are presented as well general problems in one and two dimensions are considered including a test consisting of an isotropic elastic solid and a compressible fluid in a fully coupled setting where the exact solution is available

computational complexity of real functions in this chapter we introduce the formal definitions of computational complexity of real functions we first review the definitions of computable real numbers and computable real functions and their basic properties then the oracle turing machine is introduced as the formal model for computing real functions this model allows us to define the complexity measures for computing real functions in a natural way the class of polynomial time computable real functions is then defined and several characterizations of this class will be given other complexity classes of real numbers and real functions such as np real functions and log space computable real functions will be defined in later chapters

curves that must be retraced we exhibit a polynomial time computable plane curve c that has finite length does not intersect itself and is smooth except at one endpoint but has the following property for every computable parametrization f of c and every positive integer m there is some positive length subcurve of c that f retraces at least m times in contrast every computable curve of finite length that does not intersect itself has a constant speed hence non retracing parametrization that is computable relative to the halting problem

subdivision stabilised immersed b spline finite elements for moving boundary flows an immersed finite element method is presented to compute flows with complex moving boundaries on a fixed cartesian grid the viscous incompressible fluid flow equations are discretized with b spline basis functions the two scale relation for b splines is used to implement an elegant and efficient technique to satisfy the lbb condition on non grid aligned fluid domains and at moving boundaries the boundary conditions are enforced with a consistent penalty method as originally proposed by nitsche in addition a special extrapolation technique is employed to prevent the loss of numerical stability in presence of arbitrarily small cut cells the versatility and accuracy of the proposed approach is demonstrated by means of convergence studies and comparisons with previous experimental and computational investigations

a polynomial time computable curve whose interior has a nonrecursive measure abstract a polynomial time computable simple curve is constructed such that its measure in the two dimensional plane is positive this construction is applied to prove the following two results 1 1 there exists a polynomial time computable simple closed curve in the two dimensional plane such that the measure of its interior region is a nonrecursive real number 2 2 there exists a polynomial time computable simple curve in the two dimensional plane such that its length is finite but is a nonrecursive real number

the immersed interface method for elliptic equations with discontinuous coefficients and singular sources the authors develop finite difference methods for elliptic equations of the form nabla cdot beta x nabla u x kappa x u x f x in a region omega in one or two space dimension

a non staggered grid fractional step method for time dependent incompressible navier stokes equations in curvilinear coordinates a numerical method for solving three dimensional time dependent incompressible navier stokes equations in curvilinear coordinates is presented the non staggered grid method originally developed by c m rhie and w l chow aiaaj 21 1525 1983 for steady state problems is extended to compute unsteady flows in the computational space the cartesian velocity components and the pressure are defined at the center of a control volume while the volume fluxes are defined at the mid point on their corresponding cell faces the momentum equations are integrated semi implicitly by the approximate factorization technique the intermediate velocities are interpolated onto the faces of the control volume to form the source terms of the pressure poisson equation which is solved iteratively with a multigrid method the compatibility condition of the pressure poisson equation is satisfied in the same manner as in a staggered grid method mass conservation can be satisfied to machine accuracy the pressure boundary condition is derived from the momentum equations solutions of both steady and unsteady problems including the large eddy simulation of a rotating and stratified upwelling flow in an irregular container established the favorable accuracy and efficiency of the present method

dispersion optimized quadrature rules for isogeometric analysis modified inner products their dispersion properties and optimally blended schemes abstract this paper introduces optimally blended quadrature rules for isogeometric analysis and analyzes the numerical dispersion of the resulting discretizations to quantify the approximation errors when we modify the inner products we generalize the pythagorean eigenvalue theorem of strang and fix the proposed blended quadrature rules have advantages over alternative integration rules for isogeometric analysis on uniform and non uniform meshes as well as for different polynomial orders and continuity of the basis the optimally blended schemes improve the convergence rate of the method by two orders with respect to the fully integrated galerkin method the proposed technique increases the accuracy and robustness of isogeometric analysis for wave propagation problems

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

finite element analysis on implicitly defined domains an accurate representation based on arbitrary parametric surfaces in this paper we present some novel results and ideas for robust and accurate implicit representation of geometric surfaces in finite element analysis the novel contributions of this paper are threefold 1 describe and validate a method to represent arbitrary parametric surfaces implicitly 2 represent arbitrary solids implicitly including sharp features using level sets and boolean operations 3 impose arbitrary dirichlet and neumann boundary conditions on the resulting implicitly defined boundaries the methods proposed do not require local refinement of the finite element mesh in regions of high curvature ensure the independence of the domain s volume on the mesh do not rely on boundary regularization and are well suited to methods based on fixed grids such as the extended finite element method xfem numerical examples are presented to demonstrate the robustness and effectiveness of the proposed approach and show that it is possible to achieve optimal convergence rates using a fully implicit representation of object boundaries this approach is one step in the desired direction of tying numerical simulations to computer aided design cad similarly to the isogeometric analysis paradigm

an immersed boundary method based on the gas kinetic bgk scheme for incompressible viscous flow in this paper an immersed boundary ib method based on the gas kinetic bhatnagar gross krook bgk scheme is proposed to simulate the incompressible viscous flow around stationary and moving rigid bodies in the presented ib method a set of lagrangian points represent the solid boundary and will exert force on the surrounding eulerian points which are the cell centers within the finite volume framework this force is calculated by a special iterative procedure and has an effect on the flux at the interface of cell which guarantees the fulfillment of the no slip boundary condition and entirely avoids the flow penetration the flow field is obtained from the bgk scheme with local grid refinement without complex mesh generation and transformation the present technique can be conveniently applied to simulations with complex and moving solid boundaries the second order temporal accuracy and better than first order spatial accuracy in l 2 norm are testified by the simulation of stokes first problem other three different test cases including flows around a stationary circular cylinder two circular cylinders in tandem and an oscillating circular cylinder demonstrate the good capability of the present method expanding the immersed boundary ib method to the finite volume gas kinetic bhatnagar gross krook bgk scheme the flux across the cell s interface will be influenced by the immersed boundary force the flow penetration to the immersed boundary is completely eliminated the present method can be conveniently applied to problems with moving objects

sharp interface cartesian grid method i an easily implemented technique for 3d moving boundary computations a cartesian grid method is developed for the simulation of incompressible flows around stationary and moving three dimensional immersed boundaries the embedded boundaries are represented using level sets and treated in a sharp manner without the use of source terms to represent boundary effects the narrow band distance function field in the level set boundary representation facilitates implementation of the finite difference flow solver the resulting algorithm is implemented in a straightforward manner in three dimensions and retains global second order accuracy the accuracy of the finite difference scheme is established and shown to be comparable to finite volume schemes that are considerably more difficult to implement moving boundaries are handled naturally the pressure solver is accelerated using an algebraic multigrid technique adapted to be effective in the presence of moving embedded boundaries benchmarking of the method is performed against available numerical as well as experimental results

space time discontinuous galerkin finite element method for two fluid flows a novel numerical method for two fluid flow computations is presented which combines the space time discontinuous galerkin finite element discretization with the level set method and cut cell based interface tracking the space time discontinuous galerkin stdg finite element method offers high accuracy an inherent ability to handle discontinuities and a very local stencil making it relatively easy to combine with local hp refinement the front tracking is incorporated via cut cell mesh refinement to ensure a sharp interface between the fluids to compute the interface dynamics the level set method lsm is used because of its ability to deal with merging and breakup also the lsm is easy to extend to higher dimensions small cells arising from the cut cell refinement are merged to improve the stability and performance the interface conditions are incorporated in the numerical flux at the interface and the stdg discretization ensures that the scheme is conservative as long as the numerical fluxes are conservative the numerical method is applied to one and two dimensional two fluid test problems using the euler equations

evaluation of kriging based surrogate models constructed from mesoscale computations of shock interaction with particles abstract macro scale computations of shocked particulate flows require closure laws that model the exchange of momentum energy between the fluid and particle phases closure laws are constructed in this work in the form of surrogate models derived from highly resolved mesoscale computations of shock particle interactions the mesoscale computations are performed to calculate the drag force on a cluster of particles for different values of mach number and particle volume fraction two kriging based methods viz the dynamic kriging method dkg and the modified bayesian kriging method mbkg are evaluated for their ability to construct surrogate models with sparse data i e using the least number of mesoscale simulations it is shown that if the input data is noise free the dkg method converges monotonically convergence is less robust in the presence of noise the mbkg method converges monotonically even with noisy input data and is therefore more suitable for surrogate model construction from numerical experiments this work is the first step towards a full multiscale modeling of interaction of shocked particle laden flows

numerical simulation of dendritic solidification with convection two dimensional geometry a front tracking method is presented for simulations of dendritic growth of pure substances in the presence of flow the liquid solid interface is explicitly tracked and the latent heat released during solidification is calculated using the normal temperature gradient near the interface a projection method is used to solve the navier stokes equations the no slip condition on the interface is enforced by setting the velocities in the solid phase to zero the method is validated through a comparison with an exact solution for a stefan problem a grid refinement test and a comparison with a solution obtained by a boundary integral method three sets of two dimensional simulations are presented a comparison with the simulations of beckermann et al j comput phys 154 468 1999 a study of the effect of different flow velocities and a study of the effect of the prandtl number on the growth of a group of dendrites growing together the simulations show that on the upstream side the dendrite tip velocity is increased due to the increase in the temperature gradient and the formation of side branches is promoted the flow has the opposite effect on the downstream side the results are in good qualitative agreement with published experimental results even though only the two dimensional aspects are examined here

higher order conformal decomposition fem cdfem abstract a higher order accurate finite element method is proposed which uses automatically generated meshes based on implicit level set data for the description of boundaries and interfaces in two and three dimensions the method is an alternative for fictitious domain and extended finite element methods the domain of interest is immersed in a background mesh composed by higher order elements the zero level sets are identified and meshed followed by a decomposition of the cut background elements into conforming sub elements adaptivity is a crucial ingredient of the method to guarantee the success of the mesh generation it ensures the successful decomposition of cut elements and enables improved geometry descriptions and approximations it is confirmed that higher order accurate results with optimal convergence rates are achieved with the proposed conformal decomposition finite element method cdfem

regular article an accurate cartesian grid method for viscous incompressible flows with complex immersed boundaries a cartesian grid method has been developed for simulating two dimensional unsteady viscous incompressible flows with complex immersed boundaries a finite volume method based on a second order accurate central difference scheme is used in conjunction with a two step fractional step procedure the key aspects that need to be considered in developing such a solver are imposition of boundary conditions on the immersed boundaries and accurate discretization of the governing equation in cells that are cut by these boundaries a new interpolation procedure is presented which allows systematic development of a spatial discretization scheme that preserves the second order spatial accuracy of the underlying solver the presence of immersed boundaries alters the conditioning of the linear operators and this can slow down the iterative solution of these equations the convergence is accelerated by using a preconditioned conjugate gradient method where the preconditioner takes advantage of the structured nature of the underlying mesh the accuracy and fidelity of the solver is validated by simulating a number of canonical flows and the ability of the solver to simulate flows with very complicated immersed boundaries is demonstrated

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

a parallel multifrontal algorithm and its implementation abstract in this paper we describe a multifrontal method for solving sparse systems of linear equations arising in finite element and finite difference methods the method proposed in this study is a combination of the nested dissection ordering and the frontal method it can significantly reduce the storage and computational time required by the conventional direct methods and is also a natural parallel algorithm in addition the method inherits major advantages of the frontal method which include a simple interface with finite element codes and an effective data structure so that the entire computation is performed element by element on a series of small linear systems with dense stiffness matrices the numerical implementation targets both distributed memory machines as well as conventional sequential machines its performance is tested through a series of examples

a partial differential equation approach to multidimensional extrapolation in this short note a general methodology for multidimensional extrapolation is presented the approach assumes a level set function exists which separates the region of known values from the region to be extrapolated it is shown that arbitrary orders of polynomial extrapolation can be formulated by simply solving a series of linear partial differential equations pdes examples of constant linear and quadratic extrapolation are given

multiphase dynamics in arbitrary geometries on fixed cartesian grids in this work a mixed eulerian lagrangian algorithm calledelafint eulerian lagrangian algorithm for interface tracking is developed further and applied to compute flows with solid fluid and fluid fluid interfaces the method is capable of handling fluid flows in the presence of both irregularly shaped solid boundaries and moving boundaries on a fixed cartesian grid the field equations are solved on the underlying fixed grid using a collocated variable pressure based formulation the moving boundary is tracked explicitly by the lagrangian translation of marker particles the moving boundary passes through the grid and the immersed boundary technique is used to handle its interaction with the underlying grid the internal solid boundaries are dealt with by using a cut cell technique particular attention is directed toward conservation and consistency in the vicinity of both phase boundaries the complex geometry feature has been tested for a variety of flow problems the performance of the immersed boundary representation is demonstrated in the simulation of newtonian liquid drops the combination of the two features is then employed in the simulation of motion of drops through constricted tubes the capabilities developed here can be useful for solving flow problems involving moving and stationary complex boundaries

a consistent direct discretization scheme on cartesian grids for convective and conjugate heat transfer a new discretization scheme on cartesian grids namely a consistent direct discretization scheme is proposed for solving incompressible flows with convective and conjugate heat transfer around a solid object the navier stokes and the pressure poisson equations are discretized directly even in the immediate vicinity of a solid boundary with the aid of the consistency between the face velocity and the pressure gradient from verifications in fundamental flow problems the present method is found to significantly improve the accuracy of the velocity and the wall shear stress it is also confirmed that the numerical results are less sensitive to the courant number owing to the consistency between the velocity and pressure fields the concept of the consistent direct discretization scheme is also explored for the thermal field the energy equations for the fluid and solid phases are discretized directly while satisfying the thermal relations that should be valid at their interface it takes different forms depending on the thermal boundary conditions dirichlet isothermal and neumann adiabatic iso heat flux boundary conditions for convective heat transfer and a fluid solid thermal interaction for conjugate heat transfer the validity of these discretizations is assessed by comparing the simulated results with analytical solutions for the respective thermal boundary conditions and it is confirmed that the present schemes also show high accuracy for the thermal field a significant improvement for the conjugate heat transfer problems is that the second order spatial accuracy and numerical stability are maintained even under severe conditions of near practical physical properties for the fluid and solid phases

adaptive isogeometric analysis by local h refinement with t splines isogeometric analysis based on non uniform rational b splines nurbs as basis functions preserves the exact geometry but suffers from the drawback of a rectangular grid of control points in the parameter space which renders a purely local refinement impossible this paper demonstrates how this difficulty can be overcome by using t splines instead t splines allow the introduction of so called t junctions which are related to hanging nodes in the standard fem obeying a few straightforward rules rectangular patches in the parameter space of the t splines can be subdivided and thus a local refinement becomes feasible while still preserving the exact geometry furthermore it is shown how state of the art a posteriori error estimation techniques can be combined with refinement by t splines numerical examples underline the potential of isogeometric analysis with t splines and give hints for further developments

fourier series expansion in a non orthogonal system of coordinates for the simulation of 3d dc borehole resistivity measurements abstract we describe a new method to simulate 3d borehole resistivity measurements at zero frequency dc the method combines the use of a fourier series expansion in a non orthogonal system of coordinates with an existing 2d goal oriented higher order self adaptive hp finite element algorithm the new method is suitable for simulating measurements acquired with borehole logging instruments in deviated wells it delivers high accuracy simulations and it enables a considerable reduction of the computational complexity with respect to available 3d simulators since the number of fourier modes basis functions needed to solve practical applications is limited typically below 10 furthermore numerical results indicate that the optimal 2d grid based on the 0th fourier mode also called central fourier mode can be employed to efficiently solve the final 3d problem thereby avoiding the expensive construction of optimal 3d grids specifically for a challenging through casing resistivity application we reduce the computational time from several days using a 3d simulator to just 2 h with the new method while gaining accuracy the new simulation method can be easily extended to different physical phenomena with similar geometries as those arising in the simulation of 3d borehole electrodynamics and sonic acoustics coupled with elasticity measurements in addition the method is especially suited for inversion since we demonstrate that the number of fourier modes needed for the exact representation of the materials is limited to only one the central mode for the case of borehole measurements acquired in deviated wells

implementation of regularized isogeometric boundary element methods for gradient based shape optimization in two dimensional linear elasticity the present work addresses shape sensitivity analysis and optimization in two dimensional elasticity with a regularized isogeometric boundary element method igabem non uniform rational b splines are used both for the geometry and the basis functions to discretize the regularized boundary integral equations with the advantage of tight integration of design and analysis the application of igabem in shape optimization reduces the mesh generation regeneration burden greatly the work is distinct from the previous literatures in igabem shape optimization mainly in two aspects 1 the structural and sensitivity analysis takes advantage of the regularized form of the boundary integral equations eliminating completely the need of evaluating strongly singular integrals and jump terms and their shape derivatives which were the main implementation difficulty in igabem and 2 although based on the same computer aided design cad model the mesh for structural and shape sensitivity analysis is separated from the geometrical design mesh thus achieving a balance between less design variables for efficiency and refined mesh for accuracy this technique was initially used in isogeometric finite element method and was incorporated into the present igabem implementation

variational multiscale residual based turbulence modeling for large eddy simulation of incompressible flows we present an les type variational multiscale theory of turbulence our approach derives completely from the incompressible navier stokes equations and does not employ any ad hoc devices such as eddy viscosities we tested the formulation on forced homogeneous isotropic turbulence and turbulent channel flows in the calculations we employed linear quadratic and cubic nurbs a dispersion analysis of simple model problems revealed nurbs elements to be superior to classical finite elements in approximating advective and diffusive processes which play a significant role in turbulence computations the numerical results are very good and confirm the viability of the theoretical framework

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

an implicit forcing immersed boundary method for simulating viscous flows in irregular domains abstract we present a method for solving the incompressible navier stokes equations in irregular domains these equations are discretized using finite difference method in a uniform cartesian grid stationary rigid boundaries are embedded in the cartesian grid and singular forces are applied at the rigid boundaries to impose the no slip conditions the singular forces are then distributed to the nearby cartesian grid points using the immersed boundary method in the present work the singular forces are computed implicitly by solving a small system of equations at each time step this system of equations is derived from a second order projection method the main advantage of this method is that it imposes the no slip boundary condition exactly and avoids the need for small time step to maintain stability the ability of the method to simulate viscous flows in irregular domains is demonstrated by applying to 2 dimensional flows past a circular cylinder multiple rigid obstacles and 3 dimensional flow past a sphere

efficient quadrature for nurbs based isogeometric analysis abstract we initiate the study of efficient quadrature rules for nurbs based isogeometric analysis a rule of thumb emerges the half point rule indicating that optimal rules involve a number of points roughly equal to half the number of degrees of freedom or equivalently half the number of basis functions of the space under consideration the half point rule is independent of the polynomial order of the basis efficient rules require taking into account the precise smoothness of basis functions across element boundaries several rules of practical interest are obtained and a numerical procedure for determining efficient rules is presented we compare the cost of quadrature for typical situations arising in structural mechanics and fluid dynamics the new rules represent improvements over those used previously in isogeometric analysis

a two dimensional conservation laws scheme for compressible flows with moving boundaries the generalized riemann problem grp scheme for the hydrodynamic conservation laws is extended to a two dimensional moving boundary tracking mbt configuration aimed at treating time dependent compressible flows with moving impermeable boundary surfaces a strang type operator splitting is employed for the integration of the conservation laws the boundary motion is also split into cartesian components that are taken with the respective operator split integration phases the conservation laws in boundary cells are integrated by a finite volume scheme that accounts for changing mesh geometry cell volume and side area through which mass momentum and energy fluxes are taken the central feature of the scheme is the algorithms for evaluating the changing mesh geometry at boundary cells these algorithms are based on a decomposition rule for evaluating polygon intersection area which has been used extensively for rezoning in hydrocodes over the past decades the decomposition rule is combined with the cartesian splitting of the boundary motion producing algorithms for the integration of conservation laws in boundary cells that are both consistent and simple consistency is taken to mean that the scheme produces an exact solution in the case of a uniform motion of the fluid and the boundary common velocity an illustrative example of shock lifting of a light weight cylinder is presented

computational complexity of two dimensional regions the computational complexity of bounded sets of the two dimensional plane is studied in the discrete computational model we introduce four notions of polynomial time computable sets in r 2 and study their relationship the computational complexity of the winding number problem membership problem distance problem and area problem is characterized by the relations between discrete complexity classes of the np theory

the p version of the finite element method in the p version of the finite element method the triangulation is fixed and the degree p of the piecewise polynomial approximation is progressively increased until some desired level of precision is reached in this paper we first establish the basic approximation properties of some spaces of piecewise polynomials defined on a finite element triangulation these properties lead to an a priori estimate of the asymptotic rate of convergence of the p version the estimate shows that the p version gives results which are not worse than those obtained by the conventional finite element method called the h version in which h represents the maximum diameter of the elements when quasi uniform triangulations are employed and the basis for comparison is the number of degrees of freedom furthermore in the case of a singularity problem we show under conditions which are usually satisfied in practice that the rate of convergence of the p version is twice that of the h version with quasi uniform mesh inve

a simple multigrid scheme for solving the poisson equation with arbitrary domain boundaries we present a new multigrid scheme for solving the poisson equation with dirichlet boundary conditions on a cartesian grid with irregular domain boundaries this scheme was developed in the context of the adaptive mesh refinement amr schemes based on a graded octree data structure the poisson equation is solved on a level by level basis using a one way interface scheme in which boundary conditions are interpolated from the previous coarser level solution such a scheme is particularly well suited for self gravitating astrophysical flows requiring an adaptive time stepping strategy by constructing a multigrid hierarchy covering the active cells of each amr level we have designed a memory efficient algorithm that can benefit fully from the multigrid acceleration we present a simple method for capturing the boundary conditions across the multigrid hierarchy based on a second order accurate reconstruction of the boundaries of the multigrid levels in case of very complex boundaries small scale features become smaller than the discretization cell size of coarse multigrid levels and convergence problems arise we propose a simple solution to address these issues using our scheme the convergence rate usually depends on the grid size for complex grids but good linear convergence is maintained the proposed method was successfully implemented on distributed memory architectures in the ramses code for which we present and discuss convergence and accuracy properties as well as timing performances

the cost of continuity a study of the performance of isogeometric finite elements using direct solvers we study the performance of direct solvers on linear systems of equations resulting from isogeometric analysis the problem of choice is the canonical laplace equation in three dimensions from this study we conclude that for a fixed number of unknowns and polynomial degree of approximation a higher degree of continuity k drastically increases the cpu time and ram needed to solve the problem when using a direct solver this paper presents numerical results detailing the phenomenon as well as a theoretical analysis that explains the underlying cause

a fixed grid sharp interface method for bubble dynamics and phase change abstract a numerical method has been developed for direct simulation of bubble dynamics with large liquid to vapor density ratio and phase change the numerical techniques are based on a fixed grid finite volume method capable of treating the interface as a sharp discontinuity the unsteady axisymmetric navier stokes equations and energy equation in both liquid and vapor phases are computed the mass momentum and energy conditions are explicitly matched at the phase boundary to determine the interface shape and movement the cubic b spline is used in conjunction with a fairing algorithm to yield smooth and accurate information of curvatures nondimensional parameters including reynolds weber and jakob numbers are varied to offer insight into the physical and numerical characteristics of the bubble dynamics based on the present sharp interface approach bubble dynamics for density ratio of 1600 or higher with and without phase change can be successfully computed

an efficient conservative cut cell method for rigid bodies interacting with viscous compressible flows a cartesian cut cell method for viscous flows interacting with freely moving boundaries is presented the method enables a sharp resolution of the embedded boundaries and strictly conserves mass momentum and energy a new explicit runge kutta scheme pc rk is introduced by which the overall computational time is reduced by a factor of up to 2 5 the new scheme is a predictor corrector type reformulation of a popular class of runge kutta methods which substantially reduces the computational effort for tracking the moving boundaries and subsequently reinitializing the solver impairing neither stability nor accuracy the structural motion is computed by an implicit scheme with good stability properties due to a strong coupling strategy and the conservative discretization of the flow solver at the material interfaces a new formulation for the treatment of small cut cells is proposed with high accuracy and robustness for arbitrary geometries based on a weighted taylor series approach solved via singular value decomposition the efficiency and the accuracy of the new method are demonstrated for several three dimensional cases of laminar and turbulent particulate flow it is shown that the new method remains fully conservative even for large displacements of the boundaries leading to a fast convergence of the fluid solid coupling while spurious force oscillations inherent to this class of methods are effectively suppressed the results substantiate the good stability and accuracy properties of the scheme even on relatively coarse meshes an efficient cartesian cut cell method for freely moving solid boundaries is presented the new predictor corrector runge kutta scheme pc rk enables simulation speedups of up to 2 5 mass momentum and energy are strictly conserved at the sharply resolved solid fluid interfaces the method exhibits very good stability and accuracy properties even for large displacements of the boundaries spurious force oscillations inherent to this class of methods are suppressed