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

constrained optimization framework for interface aware sub scale dynamics discrete closure model for multimaterial cells in lagrangian cell centered hydrodynamics abstract we present the new discrete optimization based interface aware sub scale dynamics ia ssd closure model for multimaterial cells for lagrangian cell centered hydrodynamics for the multimaterial cell the kinematic and thermodynamic properties e g velocity density pressure and internal energy will typically vary between the materials the discrete closure model is responsible for an accurate update of the thermodynamic states of the individual material components in the multimaterial cell and for determining the nodal forces that move the vertices of the cell the ia ssd closure model consists of two stages a bulk stage followed by a sub scale stage during the bulk stage the total change in the volume of the cell total force applied to the cell and total work done on the cell are distributed between the materials to update their volume velocity and total energy this distribution is performed using volume fractions of the materials 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 used 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 are 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 it is a first of its kind application of constrained optimization to develop discrete closure models in a more rigorous fashion the pair wise interaction between materials is essentially one dimensional in the direction that is normal to interface for this reason we demonstrate in this paper the performance of our new model on one dimensional numerical examples

a novel method for solving a class of singularly perturbed boundary value problems based on reproducing kernel method abstract in this paper a novel method is presented for solving a class of singularly perturbed boundary value problems firstly the original problem is reformulated as a new boundary value problem whose solution does not change rapidly via a proper transformation then the reproducing kernel method is employed to solve the boundary value new problem numerical results show that the present method can provide very accurate analytical approximate solutions

a unified coordinate system for solving the three dimensional euler equations it is well known that the use of eulerian coordinates for shock capturing methods results in badly smeared slip lines and that lagrangian coordinates while capable of producing sharp slip line resolution may result in severe grid deformation causing inaccuracy and even breakdown of computation a unified coordinate system is introduced in which the flow variables are considered to be functions of time and of some permanent identification of pseudo particles which move with velocity hq q being the velocity of fluid particles it includes the eulerian coordinates as a special case when h 0 and the lagrangian when h 1 for two dimensional inviscid flow the free function h is chosen so as to preserve the grid angles this results in a coordinate system which avoids excessive numerical diffusion across slip lines in the eulerian coordinates and avoids severe grid deformation in the lagrangian coordinates yet it retains sharp resolution of slip lines especially for steady flow furthermore the two dimensional unsteady euler equations of gasdynamics in the unified coordinates are found to be hyperbolic for all values of h except when h 1 i e lagrangian in the latter case the euler equations are only weakly hyperbolic lacking one eigenvector although all eigenvalues are real the consequences of this deficiency of the lagrangian coordinates are pointed out in connection with numerical computation

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

lagrangian gas dynamics in two dimensions and lagrangian systems we propose a new and canonical way of writing the equations of gas dynamics in lagrangian coordinates in two dimensions as a weakly hyperbolic system of conservation laws one part of the system is called the physical part and contains physical variables the other part is the geometrical part we show that the physical part is symmetrizable we show that the weak hyperbolicity is due to shear contact discontinuities free divergence constraints play an important role in the system we prove the l2 stability of the physical part of the system based on this formulation we derive a new conservative and entropy consistent finite volume numerical scheme we prove the stability of the numerical scheme numerical results show the potential interest of this approach various examples born infeld mhd 3d lagrangian gas dynamics can be written using the same abstract formalism

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

an approximate method for solving a class of singular perturbation problems abstract an approximate method for the numerical solution of a class of singularly perturbed two point boundary value problems is presented the given region is divided into inner and outer regions the original second order problem is replaced by an asymptotically equivalent first order problem and solved as an initial value problem in the inner region a terminal boundary condition is then obtained from the solution of the inner region problem in turn an outer region problem is obtained by replacing the second order differential equation by an approximate first order differential equation with a small deviating argument and solved efficiently by employing the trapezoidal formula coupled with a discrete invariant imbedding algorithm the proposed method is iterative on the terminal point of the inner region problem several numerical examples have been solved to demonstrate the applicability of the method

initial value technique for singularly perturbed two point boundary value problems using an exponentially fitted finite difference scheme in this paper we present an approximate method initial value technique for the numerical solution of quasilinear singularly perturbed two point boundary value problems in ordinary differential equations having a boundary layer at one end left or right point it is motivated by the asymptotic behavior of singular perturbation problems the original problem is reduced to an asymptotically equivalent first order initial value problem by approximating the zeroth order term by outer solution obtained by asymptotic expansion and then this initial value problem is solved by an exponentially fitted finite difference scheme some numerical examples are given to illustrate the given method it is observed that the presented method approximates the exact solution very well for crude mesh size h

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

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

algebraic limitations on two dimensional hydrodynamics simulations algebraic limitations imposed by the use of connected straightline segments to define meshes for hydrodynamics simulations in two dimensional cylindrical geometries are shown it is shown that in the simplest smooth isentropic flow of the spherical expansion of a gas with point symmetry commonly and currently used finite difference finite volume or finite element staggered grid hydrodynamics equations cannot simultaneously preserve energy entropy and sphericity on an equal angler mesh it is further shown why finite difference codes tend to preserve sphericity and entropy while finite element codes tend to preserve sphericity and energy exact difference representations of interface cell face pressures and work terms and of the elements of the strain rate tensor in a cell are shown

a cell centered lagrangian scheme for two dimensional compressible flow problems we present a new lagrangian cell centered scheme for two dimensional compressible flows the primary variables in this new scheme are cell centered i e density momentum and total energy are defined by their mean values in the cells the vertex velocities and the numerical fluxes through the cell interfaces are not computed independently contrary to standard approaches but are evaluated in a consistent manner due to an original solver located at the nodes the main new feature of the algorithm is the introduction of four pressures on each edge two for each node on each side of the edge this extra degree of freedom allows us to construct a nodal solver which fulfills two properties first the conservation of momentum and total energy is ensured second a semidiscrete entropy inequality is provided in the case of a one dimensional flow the solver reduces to the classical godunov acoustic solver it can be considered as its two dimensional generalization many numerical tests are presented they are representative test cases for compressible flows and demonstrate the robustness and the accuracy of this new solver

periodic homotopy groups of exceptional lie groups torsion free cases the v 1 periodic homotopy groups v 1 751 x p are computed explicitly for all pairs x p where x is an exceptional lie group whose integral homology has no p torsion this yields new lower bounds for p exponents of actual homotopy groups of these spaces delicate calculations with the unstable novikov spectral sequence are required in the proof

a maximum entropy method for solving the boundary value problem of second order ordinary differential equations we propose a new method to solve the boundary value problem for a class of second order linear ordinary differential equations which has a non negative solution the method applies the maximum entropy principle to approximating the solution numerically the theoretical analysis and numerical examples show that our method is convergent we propose a new method to solve the boundary value problem for a class of second order linear ordinary differential equations which has a non negative solution the method applies the maximum entropy principle to approximating the solution numerically the theoretical analysis and numerical examples show that our method is convergent

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 general topology godunov method we describe a numerical technique for solving 2 dimensional compressible multimaterial problems using a general topology mesh multimaterial problems are characterized by the presence of interfaces whose shapes may become arbitrarily complex in the course of dynamic evolution computational methods based on more conventional fixed connectivity quadrilateral meshes do not have adequate flexibility to follow convoluted interface shapes and frequently fail due to excessive mesh distortion the present method is based on a mesh of arbitrary polygonal cells because this mesh is dual to a triangulation its topology is unrestricted and it is able to accommodate arbitrary boundary shapes additionally this mesh is able to quickly and smoothly change local mesh resolution thus economizing on the number of mesh cells and it is able to improve mesh isotropy because in a region of uniform mesh the cells tend to become regular hexagons the underlying algorithms are based on those of the caveat code these consist of an explicit finite volume cell centered arbitrary lagrangian eulerian ale technique coupled with the godunov method which together are readily adaptable to a general topology mesh several special techniques have been developed for this extension to a more general mesh they include an interface propagation scheme based on huygens construction a near lagrangian mesh rezoning algorithm that minimizes advection while enhancing mesh regularity an efficient global remapping algorithm that is capable of conservatively transferring quantities from one general mesh to another and various mesh restructering algorithms such as mesh reconnection smoothing and point addition and deletion

operator splitting and time accuracy in lagrange plus remap solution methods operator splitting and time accuracy in lagrange plus remap solution methods for the hydrodynamics equations are investigated the time accuracy of the common solution approach is shown both analytically and numerically to be limited to first order due to operator splitting errors low order time integration of the remap terms and other postulated first order errors even if the lagrange step is second order accurate in time additional numerical studies are used to demonstrate how these errors can be eliminated with an unsplit treatment that solves the remap terms directly the discontinuous remap method in which a new mesh is generated during the remap step also is shown to be first order accurate in time

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

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