a tensor artificial viscosity using a mimetic finite difference algorithm abstract we have developed a two dimensional tensor artificial viscosity for finite difference shock wave computations the discrete viscosity tensor is formed by multiplying the gradient of velocity tensor by a scalar term the scalar term is based on the form of viscosity first presented by kurapatanko and also contains a limiter designed to switch off the viscosity for shockless compression and rigid body rotation mimetic discretizations are used to derive the form of the momentum and energy equations for a nonorthogonal grid where the viscosity tensor is evaluated at the zone edges the advantage of the tensor viscosity is a reduction of the dependence of the solution on the relation of the grid to the flow structure

induced i l fuzzy topological spaces abstract in this paper we introduce the notion of induced i l fuzzy topological spaces by using the i l valued lower semicontinuous mappings and investigate some of their basic properties in particular we give a necessary and sufficient condition for an i l fuzzy topological space to be induced and prove that this kind of induced space preserves the cartesian product and the n compactness as well

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

an improved 2d mof method by using high order derivatives abstract the mof moment of fluid method is one of the most accurate approaches among various interface reconstruction algorithms alike other second order methods the mof method needs to solve an implicit optimization problem to obtain the optimal approximate interface so an iteration process is inevitable under most circumstances in order to solve the optimization efficiently the properties of the objective function are worthy of studying in 2d problems the first order derivative has been deduced and applied in the previous researches in this paper the high order derivatives of the objective function are deduced on the convex polygon we show that the n th n 2 order derivatives are discontinuous and the number of the discontinuous points is two times the number of the polygon edge a rotation algorithm is proposed to successively calculate these discontinuous points thus the target interval where the optimal solution is located can be determined since the high order derivatives of the objective function are continuous in the target interval the iteration schemes based on high order derivatives can be used to improve the convergence rate moreover when iterating in the target interval the value of objective function and its derivatives can be directly updated without explicitly solving the volume conservation equation the direct update makes a further improvement of the efficiency especially when the number of edges of the polygon is increasing the halley s method which is based on the first three order derivatives is applied as the iteration scheme in this paper and the numerical results indicate that the cpu time is about half of the previous method on the quadrilateral cell and is about one sixth on the decagon cell

moment of fluid interface reconstruction method in multi material arbitrary lagrangian eulerian mmale algorithms a new lagrangian interface evolution method to use in a multi material ale mmale algorithm is proposed the input data set for the method consists of the volume fractions after the lagrangian step and the orientation position of the interfaces before this step in the present work the moment of fluid mof method is used at the end of the ale cycle to provide required interface information for this method compared to traditional reconstruction methods in addition to the volume fractions the proposed method incorporates the configuration of the interfaces in the calculations the effect of compressibility is also considered in prioritizing the materials the capability of the method is assessed by testing its potential to track interfaces through large topology change in the domains which contain two or more materials numerical results for these tough test problems provide enough evidence for the accuracy and robustness of the proposed method

a robust and efficient polyhedron subdivision and intersection algorithm for three dimensional mmale remapping the multi material arbitrary lagrangian eulerian mmale method is an effective way to simulate the multi material flow with severe surface deformation comparing with the traditional arbitrary lagrangian eulerian ale method the mmale method allows for multiple materials in a single cell which overcomes the difficulties in grid refinement process in recent decades many researches have been conducted for the lagrangian rezoning and surface reconstruction phases but less attention has been paid to the multi material remapping phase especially for the three dimensional problems due to two complex geometric problems the polyhedron subdivision and the polyhedron intersection in this paper we propose a clipping and projecting algorithm for polyhedron intersection whose basic idea comes from the commonly used method by grandy 1999 29 and jia et al 2013 34 our new algorithm solves the geometric problem by an incremental modification of the topology based on segmentplane intersections a comparison with jia et al 2013 34 shows our new method improves the efficiency by 55 to 65 when calculating polyhedron intersections moreover the instability caused by the geometric degeneracy can be thoroughly avoided because the geometry integrity is preserved in the new algorithm we also focus on the polyhedron subdivision process and describe an algorithm which could automatically and precisely tackle the various situations including convex non convex and multiple subdivisions numerical studies indicate that by using our polyhedron subdivision and intersection algorithm the volume conversation of the remapping phase can be exactly preserved in the mmale simulation

3d unstructured mesh ale hydrodynamics with the upwind discontinuous finite element method abstract we describe a numerical scheme to solve 3d arbitrary lagrangian eulerian ale hydrodynamics on an unstructured mesh using discontinuous finite element space and an explicit runge kutta time discretization this scheme combines the accuracy of a higher order godunov scheme with the unstructured mesh capabilities of finite elements that can be explicitly evolved in time the spatial discretization uses trilinear isoparametric elements tetrahedrons pyramids prisms and hexahedrons in which the primitive variables mass density velocity and pressure are piecewise trilinear upwinding is achieved by using roe s characteristic decomposition of the inter element boundary flux depending on the sign of characteristic wave speeds the characteristics are evaluated at the roe average of variables on both sides of the inter element boundary for a general equation of state an explicit second order runge kutta time stepping is used for the time discretization to capture shocks we have generalized van leer s 1d nonlinear minmod slope limiter to 3d using a quadratic programming scheme for very strong shocks we find it necessary to supplement this with a godunov stabilization where the trilinear representation of the variables is reduced to its constant average value the resulting numerical scheme has been tested on a variety of problems relevant to icf inertial confinement fusion target design and appears to be robust it accurately captures shocks and contact discontinuities without unstable oscillations and has second order accuracy in smooth regions object oriented programming with the c programming language was used to implement our numerical scheme the object oriented design allows us to remove the complexities of an unstructured mesh from the basic physics modules and thereby enables efficient code development

on r l o we prove there exists a somewhat simpler description of l and its fuzzy topology than the one given by hutton 2 and a considerably simpler description of than the one given by rodabaugh 4 as a result of this simplification we are able to give a direct proof of the continuity of an open problem in 4

hodge tate periods and p adic abelian integrals the object of this paper is to present two results the first is a new description of one of the maps involved in the hodge tate decomposit ion of an abelian variety with good reduction this result complements recent work of fontaine if and combined with it yields a new proof of this decomposition the second result is a limit formula involving the integrals described in c for what may properly be called hodge tate periods we will now describe in more detail the contents of this paper the notations used below are standard and are also explained in sect i let p be a fixed rational prime let ii p be the field of p adic numbers and p the completion of a fixed algebraic closure of ii p let a be an abelian variety defined over a complete discretely valued subfield k of cp let i denote the dual abelian variety and t a the p tate module of a set v a t a 1igp as a g autr the theorem of tate and raynaud which describes the decomposit ion of v a is equivalent to the following assertion there exist canonical g equivariant 2flinear maps

infinite sum on the fuzzy real line the fuzzy addition on the fuzzy real line due to hutton 1975 has been investigated by lowen 1983 rodabaugh 1982 and wang 1988 in this paper we introduce the notion of infinite sums on the fuzzy real line give a necessary and sufficient condition for an infinite sum to be convergent and discuss some properties of the cut of infinite sums

stable time step estimation for multi material eulerian hydrocodes the time step estimates used in explicit multi material eulerian hydrocodes are based on a series of upper bound approximations that can lead to unnecessarily small time steps since the cost of the calculation is proportional to the number of time steps the underestimation of the time step size has a direct adverse effect on the cost an accurate method for calculating the stable time step size is presented with example calculations

contact in a multi material eulerian finite element formulation in a multi material eulerian formulation for problems in solid mechanics the default interaction between the surfaces of adjacent materials is perfect bonding in contrast for the lagrangian formulation the surfaces don t interact and may pass through each other freely unless a contact algorithm is introduced to enforce the contact inequality constraints sophisticated contact algorithms have been developed for lagrangian calculations and are routinely used in large deformation applications such as automobile crashworthiness in comparison the eulerian finite element method lags far behind the lagrangian finite element method in modeling contact a natural application for the eulerian formulation is machining which includes large deformations material failure and contact of a tool with a workpiece making its simulation a challenging problem the simulation of the isolated chips involving material failure due to shear bands is used as a test problem to investigate the strength and weakness of the different approaches to implementing contact in multi material eulerian calculations

weak consistency of the cell centered lagrangian glace scheme on general meshes in any dimension abstract we study the consistency in the weak sense of the cell centered lagrangian scheme glace the main result is that glace is weakly consistent on general meshes in any dimension the proof relies on a new formula for some geometric vectors defined at the corners of the cell and which are basic lagrangian objects it validates theoretically the use of isoparametric elements based on q 1 integration for 3d lagrangian compressible gas dynamics calculations finally we give the result of a simple convergence test in 2d

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

a mixture theory for contact in multi material eulerian formulations abstract multi material eulerian methods are attractive for solving a broad range of highly nonlinear problems in solid mechanics because they permit arbitrarily large deformations and automatically allow new free surfaces to evolve many of these problems involve the contact and separation of surfaces there is an extensive literature for including contact inequality constraints in lagrangian finite element methods but little consideration has been given to contact in eulerian methods the contact forces between adjacent bodies are governed by the mixture theory in the eulerian formulation making the mixture theory the natural starting point in the treatment of the contact inequalities a mixture theory that equilibrates the traction across the material interface and limits the shear stress according to a coulomb friction law is developed

a meshfree contact detection algorithm a new contact detection algorithm is proposed which is based on the moment matrix of the meshfree interpolation this new contact detection algorithm has several advantages over conventional contact detection algorithms they are ease of implementation accuracy in detection it is scalable in parallel computations and applicable to a wide variety of problems because the moment matrix is computed in constructing meshfree approximation the algorithm is very suitable for meshfree computations though it can be used in finite element computations as well moreover such contact detection algorithm can be conveniently implemented to detect self contact both algorithmic and mathematical aspects of the contact algorithm are discussed in the paper to validate the proposed contact algorithm the taylor bar problem is computed using both meshfree method and finite element method fem to simulate a viscoplastic projectile impacting with a rigid target at high speed numerical results obtained with the meshfree contact algorithm show that this new contact algorithm can accurately predict the contact as well as separation of projectile and target in addition the numerical results obtained show distinct and detailed shear band formations occurring in the impact event

a tensor artificial viscosity using a finite element approach we derive a tensor artificial viscosity suitable for use in a 2d or 3d unstructured arbitrary lagrangian eulerian ale hydrodynamics code this work is similar in nature to that of campbell and shashkov 1 however our approach is based on a finite element discretization that is fundamentally different from the mimetic finite difference framework the finite element point of view leads to novel insights as well as improved numerical results we begin with a generalized tensor version of the von neumann richtmyer artificial viscosity then convert it to a variational formulation and apply a galerkin discretization process using high order gaussian quadrature to obtain a generalized nodal force term and corresponding zonal heating or shock entropy term this technique is modular and is therefore suitable for coupling to a traditional staggered grid discretization of the momentum and energy conservation laws however we motivate the use of such finite element approaches for discretizing each term in the euler equations we review the key properties that any artificial viscosity must possess and use these to formulate specific constraints on the total artificial viscosity force term as well as the artificial viscosity coefficient we also show that under certain simplifying assumptions the two dimensional scheme from 1 can be viewed as an under integrated version of our finite element method this equivalence holds on general distorted quadrilateral grids finally we present computational results on some standard shock hydro test problems as well as some more challenging problems indicating the advantages of the new approach with respect to symmetry preservation for shock wave propagation over general grids

using a curvilinear grid to construct symmetry preserving discretizations for lagrangian gas dynamics the goal of this paper is to construct discretizations for the equations of lagrangian gas dynamics that preserve plane cylindrical and spherical symmetry in the solution of the original differential equations the new method uses a curvilinear grid that is reconstructed from a given logically rectangular distribution of nodes the sides of the cells of the reconstructed grid can be segments of straight lines or arcs of local circles our procedure is exact for straight lines and circles that is it reproduces rectangular and polar grids exactly we use the method of support operators to construct a conservative finite difference method that we demonstrate will preserve spatial symmetries for certain choices of the initial grid we also introduce a curvilinear version of artificial edge viscosity that also preserves symmetry we present numerical examples to demonstrate our theoretical considerations and the robustness of the new method

fuzzy integers fuzzy rationals and other subspaces of the fuzzy real line abstract it is the purpose of this note to draw the attention to some noteworthy subspaces of the fuzzy real line m r which can be characterized by means of classical well known sequence spaces and function spaces

an implicit particle in cell method for granular materials an implicit in time method for granular materials is described the method combines the material point method a first order contact algorithm and a newton krylov equation solver to give improved energy conservation stabilization of the finite grid instability and the correct description of collisions between grains

space time discontinuous galerkin finite element method with dynamic grid motion for inviscid compressible flows ii efficient flux quadrature a new and efficient quadrature rule for the flux integrals arising in the space time discontinuous galerkin discretization of the euler equations in a moving and deforming space time domain is presented and analyzed the quadrature rule is a factor three more efficient than the commonly applied quadrature rule and does not affect the local truncation error and stability of the numerical scheme the local truncation error of the resulting numerical discretization is determined and is shown to be the same as when product gauss quadrature rules are used details of the approximation of the dissipation in the numerical flux are presented which render the scheme consistent and stable the method is successfully applied to the simulation of a three dimensional transonic flow over a deforming wing

the construction of compatible hydrodynamics algorithms utilizing conservation of total energy the principal goal of all numerical algorithms is to represent as faithfully and accurately as possible the underlying continuum equations to which a numerical solution is sought however in the transformation of the equations of fluid dynamics into discretized form important physical properties are either lost or obeyed only to an approximation that often becomes worse with time this is because the numerical methods used to form the discrete analog of these equations may only represent them to some order of local truncation error without explicit regard to global properties of the continuum system although a finite truncation error is inherent to all discretization methods it is possible to satisfy certain global properties such as conservation of mass momentum and total energy to numerical roundoff error the purpose of this work is to show how these equations can be differenced compatibly so that they obey the aforementioned properties in particular it is shown how conservation of total energy can be utilized as an intermediate device to achieve this goal for the equations of fluid dynamics written in lagrangian form and with a staggered spatial placement of variables for any number of dimensions and in any coordinate system for staggered spatial variables it is shown how the momentum equation and the specific internal energy equation can be derived from each other in a simple and generic manner by use of the conservation of total energy this allows for the specification of forces that can be of an arbitrary complexity such as those derived from an artificial viscosity or subzonal pressures these forces originate only in discrete form nonetheless the change in internal energy caused by them is still completely determined the procedure given here is compared to the method of support operators to which it is closely related difficulties with conservation of momentum volume and entropy are also discussed the proper treatment of boundary conditions and differencing with respect to time are detailed

consistent tangent operators for rate independent elastoplasticity abstract it is shown that for problems involving rate constitutive equations such as rate independent elastoplasticity the notion of consistency between the tangent stiffness operator and the integration algorithm employed in the solution of the incremental problem plays a crucial role in preserving the quadratic rate of asymptotic convergence of iterative solution schemes based upon newton s method within the framework of closest point projection algorithms a methodology is presented whereby tangent operators consistent with this class of algorithms may be systematically developed to wit associative j 2 flow rules with general nonlinear kinematic and isotropic hardening rules as well as a class of non associative flow rules are considered the resulting iterative solution scheme preserves the asymptotic quadratic convergence characteristic of newton s method whereas use of the socalled elastoplastic tangent in conjunction with a radial return integration algorithm a procedure often employed results in newton type of algorithms with suboptimal rate of convergence application is made to a set of numerical examples which include saturation hardening laws of exponential type

the internal consistency stability and accuracy of the discrete compatible formulation of lagrangian hydrodynamics this work explores the somewhat subtle meaning and consequences of the salient properties of the discrete compatible formulation of lagrangian hydrodynamics in particular since this formulation preserves total energy to roundoff error the amount of error in the conservation of total energy cannot be used to gauge the internal consistency of calculations as is often done with the older forms of this algorithm however the compatible formulation utilizes two definitions of zone volume the first is the usual definition whereby the volume of a zone is defined as some prescribed function of the coordinates of the points that define it the second is given as the integration in time of the continuity equation for zone volume as expressed in lagrangian form it is the use of this latter volume in the specific internal energy equation that enables total energy to be exactly conserved these two volume definitions are generally not precisely equal it is the analysis of this difference that forms the first part of this study it is shown that this difference in zone volumes can be used to construct a practical internal consistency measure that not only takes the place of the lack of total energy conservation of the older forms of lagrangian hydrodynamics but is more general in that it can be defined on a single zone basis it can also be used to ascertain the underlying spatial and temporal order of accuracy of any given set of calculations the difference in these two definitions of zone volume may be interpreted as a type of entropy error however this entropy error is found to be significant only when a given calculation becomes numerically unstable otherwise it remains at or far beneath truncation error levels in fact it can be utilized to provide an upper bound on the size of the spatial truncation error for a stable computation it is also shown how this volume difference can be used as an indicator of numerical difficulties since exact local conservation of total energy does not guarantee numerical stability or the quality of any numerical calculation the discrete compatible formulation of lagrangian hydrodynamics utilizes a two level predictor corrector type of time integration scheme a stability analysis both analytical and numerical is given this analysis reveals a novel stability diagram that has not been heretofore published and gives definitive information as to how the stabilizing corrector step should be centered in time

convergence of sequences on the fuzzy real line the infinite sum on the fuzzy real line r i has been investigated by wang geping fuzzy sets and systems 98 1998 241 in this paper we introduce the notion of convergence of sequences on r i give some conditions for a sequence on r i to be convergent and discuss some properties of convergence of sequences on r i

torsional springs for two dimensional dynamic unstructured fluid meshes abstract dynamic fluid grids are commonly used for the solution of flow problems with moving boundaries they are often represented by a network of fictitious lineal springs that can become unreliable when the fluid mesh undergoes large displacements and or deformations in this paper we propose to control the arbitrary motion of two dimensional dynamic unstructured fluid grids with additional torsional springs we show that such springs can be designed to prohibit the interpenetration of neighboring triangles and therefore to provide the method of spring analogy with the robustness needed for enlarging its range of applications we illustrate our new dynamic mesh motion algorithm with several examples that highlight its advantages in terms of robustness quality and performance

on the shafarevich and tate conjectures for hyperkahler varieties 1 problems and results 2 polarized hyperk ihler varieties and cubic fourfoids 3 the period mapping 4 the kuga satake construction 5 the kuga satake construction in the relative setting 6 proof of theorem 1 5 1 7 proof of theorem 1 6 1 8 descent 9 proof of theorem 1 3 1 appendix 1 spinorial shimura varieties appendix 2 a short proof of the hodge conjecture for cubic fourfolds 205 210 212 216 220 224 229 233 240 244 245

space time discontinuous galerkin finite element method with dynamic grid motion for inviscid compressible flows i general formulation a new space time discontinuous galerkin finite element method for the solution of the euler equations of gas dynamics in time dependent flow domains is presented the discontinuous galerkin discretization results in an efficient elementwise conservative upwind finite element method which is particularly well suited for local mesh refinement the upwind scheme uses a formulation of the hllc flux applicable to moving meshes and several formulations for the stabilization operator to ensure that monotone solutions around discontinuities are investigated the non linear equations of the space time discretization are solved using a multigrid accelerated pseudo time integration technique with an optimized runge kutta method the linear stability of the pseudo time integration method is investigated for the linear advection equation the numerical scheme is demonstrated with simulations of the flow field in a shock tube a channel with a bump and an oscillating naca 0012 airfoil these simulations show that using the data at the superconvergence points the accuracy of the numerical discretization is o h5 2 in space for smooth subsonic flows both on structured and on locally refined meshes and that the space time adaptation can significantly improve the accuracy and efficiency of the numerical method

elimination of artificial grid distortion and hourglass type motions by means of lagrangian subzonal masses and pressures the bane of lagrangian hydrodynamics calculations is the premature breakdown of grid topology that results in severe degradation of accuracy and run termination often long before the assumption of a lagrangian zonal mass has ceased to be valid at short spatial grid scales this is usually referred to by the terms hourglass mode or keystone motion associated in particular with underconstrained grids such as quadrilaterals and hexahedrons in two and three dimensions respectively at longer spatial lengths relative to the grid spacing there is what is referred to ubiquitously as spurious vorticity or the long thin zone problem in both cases the result is anomalous grid distortion and tangling that has nothing to do with the actual solution as would be the case for turbulent flow in this work we show how such motions can be eliminated by the proper use of subzonal lagrangian masses and associated densities and pressures these subzonal pressures give rise to forces that resist these spurious motions the pressure is no longer a constant in a zone it now accurately reflects the density gradients that can occur within a zone due to its differential distortion subzonal lagrangian masses can be choosen in more than one manner to obtain subzonal density and pressure variation however these masses arise in a natural way from the intersection of the lagrangian contours through which no mass flows that are associated with both the lagrangian zonal and nodal masses in a staggered spatial grid hydrodynamics formulation this is an extension of the usual lagrangian assumption that is often applied to only the zonal mass we show that with a proper discretization of the subzonal forces resulting from subzonal pressures hourglass motion and spurious vorticity can be eliminated for a very large range of problems

a robust mof method applicable to severely deformed polygonal mesh abstract the mof moment of fluid method is an accurate approach for interface reconstruction in numerical simulation of multi material fluid flow so far most works focus on improving its accuracy and efficiency such as developing analytic reconstruction method and deducing the iteration schemes based on high order derivatives of the objective function in this paper we mainly concern on improving its robustness especially for severely deformed polygonal meshes in which case the objective function has multiple minimum value points by using an efficient method for solving multiple roots of the nonlinear equation in large scope a new algorithm is developed to enhance robustness of the mof method the main idea of this algorithm is as follows the first derivative of the objective function is continuous so the minimum value points of the objective function must be the zero points of the first derivative instead of finding the zero points of the first derivative directly we turn to calculating the minimum value points also zero points of the square of the first derivative which is a convex function on a neighborhood of each zero point applying the properties of convex function the neighbor of each extreme minimum point of it can be obtained efficiently then each zero point of the square of the first derivative can be obtained using the iterative formula in its neighbor finally by comparing the values of the objective function at these zero points of the first derivative the global minimum value point of the objective function can be found and is the desired solution the new algorithm only uses the first derivative of the objective function it doesn t need an initial guess for the solution which has to be carefully chosen in previous works numerical results are presented to demonstrate the accuracy and robustness of this new algorithm the results show that it is applicable to severely deformed polygonal mesh even with concave cells

on arithmetic class invariants let f be a number field with ring of integers of and let s be a finite set of places of f assume that s contains the set s of archimedean places of f and write sf for the set of finite places contained in s let os or o when there is no danger of confusion denote the ring of sf integers of f write f for an algebraic closure of f let y be any scheme over spec o suppose thatg is a finite flat commutative group scheme over y of exponent n and let g denote the cartier dual of g let x y be ag torsor and write 0 g y for the trivialg torsor thenox is anog comodule and so it is also anogd module see 12 as an ogd module ox is locally free of rank one and it therefore gives a line bundle m over g set l m m 1 0 then the map h y g pic g l

an exact general remeshing scheme applied to physically conservative voxelization we present an exact general remeshing scheme to compute analytic integrals of polynomial functions over the intersections between convex polyhedral cells of old and new meshes in physics applications this allows one to ensure global mass momentum and energy conservation while applying higher order polynomial interpolation we elaborate on applications of our algorithm arising in the analysis of cosmological n body data computer graphics and continuum mechanics problems we focus on the particular case of remeshing tetrahedral cells onto a cartesian grid such that the volume integral of the polynomial density function given on the input mesh is guaranteed to equal the corresponding integral over the output mesh we refer to this as physically conservative voxelization at the core of our method is an algorithm for intersecting two convex polyhedra by successively clipping one against the faces of the other this algorithm is an implementation of the ideas presented abstractly by sugihara 48 who suggests using the planar graph representations of convex polyhedra to ensure topological consistency of the output this makes our implementation robust to geometric degeneracy in the input we employ a simplicial decomposition to calculate moment integrals up to quadratic order over the resulting intersection domain we also address practical issues arising in a software implementation including numerical stability in geometric calculations management of cancellation errors and extension to two dimensions in a comparison to recent work we show substantial performance gains we provide a c implementation intended to be a fast accurate and robust tool for geometric calculations on polyhedral mesh elements

momentum advection on unstructured staggered quadrilateral meshes the finite element method as applied to problems in solid mechanics typically uses a mesh with the velocities at the nodes and the remaining solution variables including the density located at the integration points the arbitrary lagrangian eulerian formulations used in solid mechanics are therefore faced with the challenge of transporting momentum which is defined in terms of variables located at separate points in space in a conservative manner two types of momentum transport methods have been developed over the years the first constructs a dual mesh with the nodes as the integration points a difficult task on an unstructured finite element mesh the second uses the original mesh and constructs auxiliary variables for transport from which the final velocity may be recovered an analysis demonstrates how the two methods are related simplified implementations of each type dual mesh and element centered are developed in detail and their performance is compared to verify the analysis copyright 2008 john wiley sons ltd

a subcell remapping method on staggered polygonal grids for arbitrary lagrangian eulerian methods we describe a new remapping algorithm for use in arbitrary lagrangian eulerian ale simulations the new features of this remapper are designed to complement a staggered mesh lagrangian phase in which the cells may be general polygons in two dimensions and which uses subcell discretizations to control unphysical mesh distortion and hourglassing our new remapping algorithm consists of three stages a gathering stage in which we interpolate momentum internal energy and kinetic energy to the subcells in a conservative way a subcell remapping stage in which we conservatively remap mass momentum internal and kinetic energy from the subcells of the lagrangian mesh to the subcells of the new rezoned mesh a scattering stage in which we conservatively recover the primary variables subcell density nodal velocity and cell centered specific internal energy on the new rezoned mesh we prove that our new remapping algorithm is conservative reversible and satisfies the debar consistency condition we also demonstrate computationally that our new remapping method is robust and accurate for a series of test problems in one and two dimensions

the force work differencing of exceptional points in the discrete compatible formulation of lagrangian hydrodynamics this study presents the force and mass discretization of exceptional points in the compatible formulation of lagrangian hydrodynamics it concludes a series of papers that develop various aspects of the theoretical exposition and the operational implementation of this numerical algorithm exceptional points are grid points at the termination of lines internal to the computational domain and where boundary conditions are therefore not applied these points occur naturally in most applications in order to ameliorate spatial grid anisotropy and the consequent timestep reduction that will otherwise arise for grids with highly tapered regions or a center of convergence they have their velocity enslaved to that of neighboring points in order to prevent large excursions of the numerical solution about them how this problem is treated is given herein for the aforementioned numerical algorithm such that its salient conservation properties are retained in doing so the subtle aspects of this algorithm that are due to the interleaving of spatial contours that occur with the use of a spatially staggered grid mesh are illuminated these contours are utilized to define both forces and the work done by them and are the central construct of this type of finite volume differencing additionally difficulties that occur due to uncertainties in the specification of the artificial viscosity are explored and point to the need for further research in this area

a two dimensional unstructured cell centered multi material ale scheme using vof interface reconstruction we present a new cell centered multi material arbitrary lagrangian eulerian ale scheme to solve the compressible gas dynamics equations on two dimensional unstructured grid our ale method is of the explicit time marching lagrange plus remap type namely it involves the following three phases a lagrangian phase wherein the flow is advanced using a cell centered scheme a rezone phase in which the nodes of the computational grid are moved to more optimal positions a cell centered remap phase which consists of interpolating conservatively the lagrangian solution onto the rezoned grid the multi material modeling utilizes either concentration equations for miscible fluids or the volume of fluid vof capability with interface reconstruction for immiscible fluids the main original feature of this ale scheme lies in the introduction of a new mesh relaxation procedure which keeps the rezoned grid as close as possible to the lagrangian one in this formalism the rezoned grid is defined as a convex combination between the lagrangian grid and the grid resulting from condition number smoothing this convex combination is constructed through the use of a scalar parameter which is a scalar function of the invariants of the cauchy green tensor over the lagrangian phase regarding the cell centered remap phase we employ two classical methods based on a partition of the rezoned cell in terms of its overlap with the lagrangian cells the first one is a simplified swept face based method whereas the second one is a cell intersection based method our multi material ale methodology is assessed through several demanding two dimensional tests the corresponding numerical results provide a clear evidence of the robustness and the accuracy of this new scheme

a compatible energy and symmetry preserving lagrangian hydrodynamics algorithm in three dimensional cartesian geometry this work presents a numerical algorithm for the solution of fluid dynamics problems with moderate to high speed flow in three dimensions cartesian geometry is chosen owing to the fact that in this coordinate system no curvature terms are present that break the conservation law structure of the fluid equations written in lagrangian form these equations are discretized utilizing compatible control volume differencing with a staggered grid placement of the spatial variables the concept of compatibility means that the forces used in the momentum equation to advance velocity are also incorporated into the internal energy equation so that these equations together define the total energy as a quantity that is exactly conserved in time in discrete form multiple pressures are utilized in each zone they produce forces that resist spurious vorticity generation this difficulty can severely limit the utility of the lagrangian formulation in two dimensions and make this representation otherwise virtually useless in three dimensions an edge centered artificial viscosity whose magnitude is regulated by local velocity gradients is used to capture shocks the particular difficulty of exactly preserving one dimensional spherical symmetry in three dimensional geometry is solved this problem has both practical and pedagogical significance the algorithm is suitable for both structured and unstructured grids limitations that symmetry preservation imposes on the latter type of grids are delineated

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

convergence in the i fuzzy real line infinite sums and sequences in the fuzzy real line have been investigated by wang in this paper a necessary and sufficient condition for a sequence in to be convergent is given in terms of cut set it is proved that fuzzy convergence of sequences in is equivalent to weak convergence as an application convergence of sequences of fuzzy numbers in the sense of chang and zadeh is discussed

constrained optimization framework for interface aware sub scale dynamics closure model for multimaterial cells in lagrangian and arbitrary lagrangian eulerian hydrodynamics abstract a systematic description of the new interface aware sub scale dynamics ia ssd closure model for the lagrangian stage of multimaterial arbitrary lagrangian eulerian methods is presented the ia ssd closure model consists of two stages during the first bulk stage the well known equal compressibility model is used during the second stage sub scale interactions of the materials inside the multimaterial cell are taken into account at this stage information about the topology of the materials inside the multimaterial cell is utilized allowing the orientations of internal interfaces to be included in the model each material interacts in a pair wise fashion with the materials with which it has a common boundary the interactions are based on the solution of the acoustic riemann problem between each pair of materials and is limited using physically justified constraints positivity of volume positivity of internal energy and controlled rate of pressure relaxation to determine the values of the limiter coefficients a constrained optimization framework is employed using a quadratic objective function with linear constraints the algorithm guarantees the positivity of the material volume and internal energy as well as the smooth relaxation of the pressure this allows a significant increase in the robustness of the overall algorithm the results of comprehensive testing of the new model have been presented for one and two dimensional multimaterial lagrangian hydrodynamics along with representative results for 2d multimaterial arbitrary lagrangian eulerian ale calculations the numerical tests have shown that in most cases the new ia ssd closure model produces better results compared to the well known tipton s closure model

fuzzy addition in the lowen fuzzy real line for l a complete chain with order reversing involution rodabaugh 7 defined a fuzzy addition in the hutton fuzzy real line r l and studied some of its properties subsequently lowen 3 made a considerably simpler description of and gave a direct proof of the continuity of in this paper we introduce the fuzzy addition in the lowen fuzzy real line r 6 we give several equivalent descriptions of which may be used for investigating some properties of moreover we prove the closedness of in the four important subspaces a r f r z r q r 4 of r this implies in particular that the sum of two fuzzy rational numbers is fuzzy rational

two new three dimensional contact algorithms for staggered lagrangian hydrodynamics abstract this paper presents two new three dimensional contact algorithms for staggered lagrangian hydrodynamics named discrete accurate matching method and discrete lagrangian multiplier method these new contact algorithms utilize the unified contact algorithm for contact searching contact segments from all potential contact surfaces are treated as a single set the basic idea of these new methods is to partition all contact segments into triangular facets firstly then for each pair of two triangular facets which correspond to a hitting node and a target node respectively these two triangular facets are projected to one plane and their intersection area is calculated the nodal masses and nodal forces of the hitting node and the target node are distributed to the intersection portions of these two triangular facets respectively according to the proportion of the intersection area to corresponding nodal area in the discrete accurate matching method the masses and forces of the intersection portions of these two triangular facets are added to corresponding contact nodes of each other and then the accelerations and velocities of the contact nodes are updated while in the discrete lagrangian multiplier method the intersection portions of these two triangular facets are considered as a 1d contact pair in the normal direction for which the contact force is explicitly calculated by the lagrangian multiplier method for a 1d contact pair of one hitting point and one target point then the contact force being added to corresponding contact nodes these new contact algorithms are assessed through several numerical tests performed on three dimensional structured and unstructured meshes the results of these tests show the accuracy and robustness of these new methods

a new exceptional points method with application to cell centered lagrangian schemes and curved meshes the aim of this paper is the numerical simulation of compressible hydrodynamic strong implosions which take place for instance in inertial confinement fusion it focuses in particular on two crucial issues for such calculations very low cfl number at the implosion center and approximation error on the initial geometry thus we propose an exceptional points method which is translation invariant and suitable to curved meshes this method is designed for cell centered lagrangian schemes glace 8 13 14 eucclhyd 25 29 several numerical examples on significant test cases are presented to show the relevance of our approach

an eulerian shell formulation for fluid structure interaction abstract coupled eulerian lagrangian formulations are currently used to analyze fluid structure interactions extending these formulations to handle large structureal deformations structural failure and perforation is challenging an eulerian structural formulation is attractive because it permits arbitrarily large deformations and permits the new surfaces associated with perforation to evolve without specialized algorithms the theoretical basis for an eulerian formulation for shells is developed and in particular the choice of a formulation for the rotations is discussed example calculations with large deformations shell thinning and weld failure are presented

analysis of a contact problem involving thermoelastic mixtures abstract in this work we study a dynamic contact problem between a thermoelastic mixture and a deformable obstacle the classical normal compliance condition is used for modeling the contact the variational formulation of this problem is written as a nonlinear coupled system of three parabolic variational equations an existence and uniqueness result is proved using the faedo galerkin method then fully discrete approximations are introduced by using the finite element method and the implicit euler scheme a discrete stability property and a priori error estimates are proved from which the decay of the discrete energy and the linear convergence of the algorithm are deduced finally some numerical simulations are presented to show the convergence and the behavior of the solution

an improved 3d mof method based on analytical partial derivatives abstract mof moment of fluid method is one of the most accurate approaches among various surface reconstruction algorithms as other second order methods mof method needs to solve an implicit optimization problem to obtain the optimal approximate surface therefore the partial derivatives of the objective function have to be involved during the iteration for efficiency and accuracy however to the best of our knowledge the derivatives are currently estimated numerically by finite difference approximation because it is very difficult to obtain the analytical derivatives of the object function for an implicit optimization problem employing numerical derivatives in an iteration not only increase the computational cost but also deteriorate the convergence rate and robustness of the iteration due to their numerical error in this paper the analytical first order partial derivatives of the objective function are deduced for 3d problems the analytical derivatives can be calculated accurately so they are incorporated into the mof method to improve its accuracy efficiency and robustness numerical studies show that by using the analytical derivatives the iterations are converged in all mixed cells with the efficiency improvement of 3 to 4 times

a lagrangian staggered grid godunov like approach for hydrodynamics much research in lagrangian staggered grid hydrodynamics sgh has focused on explicit viscosity models for adding dissipation to a calculation that has shocks the explicit viscosity is commonly called artificial viscosity recently researchers have developed hydrodynamic algorithms that incorporate approximate riemann solutions on the dual grid 28 29 35 30 2 3 this approach adds dissipation to the calculation via solving a riemann like problem in this work we follow the works of 28 29 35 30 and solve a multidirectional riemann like problem at the cell center the riemann like solution at the cell center is used in the momentum and energy equations the multidirectional riemann like problem used in this work differs from previous work in that it is an extension of the cell centered hydrodynamics cch nodal solution approach in 7 incorporating the multidirectional riemann like problem from 7 into sgh has merits such as the ability to resist mesh instabilities like hourglass null modes and chevron null modes the approach is valid for complex multidimensional flows with strong shocks numerical details and test problems are presented

a conservative nodal variational multiscale method for lagrangian shock hydrodynamics abstract a new method based on a continuous piece wise linear approximation of the equations for lagrangian shock hydrodynamics is presented numerical instabilities are controlled by a stabilizing operator derived using the paradigm of the variational multiscale analysis encouraging numerical comparisons with existing methods in the case of quadrilateral and hexahedral elements indicate that the proposed method is capable of preventing hourglass patterns in the solution while maintaining accuracy in regions of smooth flow the proposed approach satisfies galilean invariance properties and hinges upon the interpretation of the lagrangian shock hydrodynamics equations as a system of nonlinear wave equations a specific implementation in terms of a predictor multi corrector version of the mid point time integrator guarantees global conservation of mass momentum and total energy for each iterate stability and formal order of accuracy are investigated applying the von neumann analysis to the linearized shock hydrodynamics equations in one dimension this approach yields tight bounds for stable time step estimation formal second order accuracy of the method in time and space and valuable indications on the choice of the most appropriate values for the stabilization parameters present in the formulation

the ball vertex method a new simple spring analogy method for unstructured dynamic meshes we study the problem of deforming unstructured grids using pseudo structural lumped parameter systems a basic network of edge springs is complemented with an additional set of linear springs that oppose element collapsing this is here achieved by confining each mesh vertex to its ball through linear springs that are attached to the vertex and to its projections on the ball entities the resulting linear pseudo structural problem can be solved efficiently with an iterative method the proposed procedures are compared with a revisited version of the torsional spring method with the help of two and three dimensional example problems

an extended eulerian method for contacts in godunov formulations summary this paper offers a new method to treat contacts in eulerian godunov formulations the inadequacy of classic eulerian formulations to describe contacts behavior is identified in the lack of degrees of freedom this limitation is overcome by employing a set of state variables per material for the calculation of multi material elements and faces the extra state variables allow for the independent analysis of the materials present in the problem materials interactions are characterized by introducing contact constraints in the formulation the locality of the method limits the memory burden copyright 2012 john wiley sons ltd

formulations of artificial viscosity for multi dimensional shock wave computations in this paper we present a new formulation of the artificial viscosity concept physical arguments for the origins of this term are given and a set of criteria that any proper functional form of the artificial viscosity should satisfy is enumerated the first important property is that by definition a viscosity must always be dissipative transferring kinetic energy into internal energy and must never act as a false pressure the artificial viscous force should be galilean invariant and vary continuously as a function of the criterion used to determine compression and expansion and remain zero for the latter case these requirements significantly constrain the functional form that the artificial viscous force can have in addition an artificial viscosity should be able to distinguish between shock wave and adiabatic compression and not result in spurious entropy production when only the latter is present it must therefore turn off completely for self similiar motion where only a uniform stretching and or a rigid rotation occurs an additional important but more subtle condition where the artificial viscosity should produce no effect is along the direction tangential to a convergent shock front since the velocity is only discontinuous in the normal direction our principal result is the development of a new formulation of an edge centered artificial viscosity that is to be used in conjunction with a staggered spatial placement of variables that meets all of these standards and without the need for problem dependent numerical coefficients that have in the past made the artificial viscosity method appear somewhat arbitrary our formulation and numerical results are given with respect to two spatial dimensions but all of our arguments carry over directly to three dimensions a central feature of our development is the implementation of simple advection limiters in a straightforward manner in more than one dimension to turn off the artificial viscosity for the above mentioned conditions and to substantially reduce its effect when strong velocity gradients are absent

numerical preservation of symmetry properties of continuum problems we investigate the problem of perfectly preserving a symmetry associated naturally with one coordinate system when calculated in a different coordinate system this allows a much wider range of problems that may be viewed as perturbations of the given symmetry to be investigated we study the problem of preserving cylindrical symmetry in two dimensional cartesian geometry and spherical symmetry in two dimensional cylindrical geometry we show that this can be achieved by a simple modification of the gradient operator used to compute the force in a staggered grid lagrangian hydrodynamics algorithm in the absence of the supposed symmetry we show that the new operator produces almost no change in the results because it is always close to the original gradient operator our technique thus results in a subtle manipulation of the spatial truncation error in favor of the assumed symmetry but only to the extent that it is naturally present in the physical situation this not only extends the range of previous algorithms and the use of new ones for these studies but for spherical or cylindrical calculations it reduces the sensitivity of the results to grid setup with equal angular zoning that has heretofore been necessary with these problems

stabilization of cell centered compressible lagrangian methods using subzonal entropy this work deals with the stabilization of early stages of lagrangian calculations of compressible gas dynamics in the context of cell centered discretization we describe a totally conservative algorithm based on an entropy analysis of the distortion of a lagrangian mesh it prevents the tangling of the mesh while remaining consistent and conservative in mass momentum and total energy the method described can be applied to any cell centered lagrangian scheme in this article we detail the extension to the cell centered glace scheme published in carre et al 2009 g carre s del pino b despres e labourasse a cell centered lagrangian hydrodynamics scheme on general unstructured meshes in arbitrary dimension j comput phys 228 2009 5160 5183 numerical tests are proposed to assess the accuracy and robustness

vorticity errors in multidimensional lagrangian codes abstract we investigate the apparent paradox as exemplified by the well known saltzman test problem of multidimensional lagrangian codes experiencing mesh tangling when computing one dimensional irrotational flows we demonstrate that the cause is the generation of spurious vorticity or vorticity error by a nonuniform mesh based on this we investigate two methods of constructing improved lagrangian vertex velocities by removing or filtering out this spurious vorticity rather than by the more common practice of introducing artificial viscosity the first method reconstructs the velocity from the known flow divergence and from the true vorticity computed by means of a transport equation the second method which is much simpler and more efficient subtracts a divergence free correction from the velocity such that the resulting velocity possesses the correct vorticity we then successfully apply this method to solve a two dimensional shock refraction problem a problem which exhibits nonzero intrinsic vorticity

a predicted newton s method for solving the interface positioning equation in the mof method on general polyhedrons abstract solving the interface positioning equation is the key procedure of the plic vof methods most of previous research only focused on the planar constant calculation and paid less attention to the relationship between the planar constant and the approximate interface orientation the latter issue is important especially for the second order iteration based plic vof method such as the mof and lvira method in these methods the most accurate interface orientation is calculated through an iterative procedure so the interface positioning equation has to be solved multiple times for the given volume fraction with different interface orientations in this situation if the incremental relation between the planar constant and the interface orientation is known a predicted planar constant can be estimated in this paper we deduce the analytical partial derivatives of the planar constant with respect to the interface orientation and use them to predict the planar constant a predicted newton s method is proposed to solve the interface positioning equation which takes the predicted planar constant as the initial guess a great deal of numerical tests are also presented in this paper to verify the robustness of the new scheme the efficiency of the proposed predicted newton s method is compared with the commonly used secant bisection method by ahn and shashkov and the numerical results indicate that the new method can reduce the iteration steps by 60 66 in solving the interface positioning equation and reduce the cpu time by 32 39 when implemented in the mof method

fuzzy addition in the l fuzzy real line abstract under the hypothesis l is a chain we construct a binary operation on the l fuzzy real line r l which reduces to the usual addition on r if is restricted to the embedded image of r in r l which yields a partially ordered abelian cancellation semigroup with identity and which is jointly fuzzy continuous on r l we show is unique i e it is the only extension of addition to r l which is consistent we study the relationship between and other fuzzy continuous extensions of the usual addition we also show that fuzzy translation is a weak fuzzy homeomorphism and under certain conditions a fuzzy homeomorphism