Lagrangian cell-centered conservative scheme

This paper presents a Lagrangian cell-centered conservative gas dynamics scheme. The piecewise constant pressures of cells arising from the current time sub-cell densities and the current time isentropic speed of sound are introduced. Multipling the initial cell density by the initial sub-cell volumes obtains the sub-cell Lagrangian masses, and dividing the masses by the current time sub-cell volumes gets the current time sub- cell densities. By the current time piecewise constant pressures of cells, a scheme that conserves the momentum and total energy is constructed. The vertex velocities and the numerical fluxes through the cell interfaces are computed in a consistent manner due to an original solver located at the nodes. The numerical tests are presented, which are representative for compressible flows and demonstrate the robustness and accuracy of the Lagrangian cell-centered conservative scheme.

High-order Lagrangian cell-centered conservative scheme on unstructured meshes

A high-order Lagrangian cell-centered conservative gas dynamics scheme is presented on unstructured meshes. A high-order piecewise pressure of the cell is intro- duced. With the high-order piecewise pressure of the cell, the high-order spatial discretiza- tion fluxes are constructed. The time discretization of the spatial fluxes is performed by means of the Taylor expansions of the spatial discretization fluxes. The vertex velocities are evaluated in a consistent manner due to an original solver located at the nodes by means of momentum conservation. Many numerical tests are presented to demonstrate the robustness and the accuracy of the scheme.

A One-Dimensional Second-Order Cell-Centered Lagrangian Scheme Satisfying the Entropy Condition

The numerical solutions of gas dynamics equations have to be consistent with the second law of thermodynamics,which is termed entropy condition.However,most cell-centered Lagrangian(CL)schemes do not satisfy the entropy condition.Until 2020,for one-dimensional gas dynamics equations,the first-order CL scheme with the hybridized flux developed by combining the acoustic approximate(AA)flux and the entropy conservative(EC)flux developed by Maire et al.was used.This hybridized CL scheme satisfies the entropy condition;however,it is under-entropic in the part zones of rarefaction waves.Moreover,the EC flux may result in nonphysical numerical oscillations in simulating strong rarefaction waves.Another disadvantage of this scheme is that it is of only first-order accuracy.In this paper,we firstly construct a modified entropy conservative(MEC)flux which can damp effectively numerical oscillations in simulating strong rarefaction waves.Then we design a new hybridized CL scheme satisfying the entropy condition for one-dimensional complex flows.This new hybridized CL scheme is a combination of the AA flux and the MEC flux.In order to prevent the specific entropy of the hybridized CL scheme from being under-entropic,we propose using the third-order TVD-type Runge-Kutta time discretization method.Based on the new hybridized flux,we develop the second-order CL scheme that satisfies the entropy condition.Finally,the characteristics of our new CL scheme using the improved hybridized flux are demonstrated through several numerical examples.

A Cell-Centered Lagrangian Scheme with an Elastic-Perfectly Plastic Solid Riemann Solver for Wave Propagations in Solids

A cell-centered Lagrangian scheme is developed for the numerical simula-tion of wave propagations in one dimensional(1D)elastic-plasticflow.The classical elastic-plastic material model initially proposed by Wilkins is adopted.The linear elas-tic model(Hooke’s Law),perfectly plastic model and von Mises yield criterion are used to describe the constitutive relationship of elastic-plastic solid.The second-order ex-tension of this scheme is achieved by a linear reconstruction method.Various numer-ical tests are simulated to check the capability of this scheme in capturing nonlinear elastic-plastic waves.Compared with the well-developed operator splitting method used in simulating elastic-plasticflow,this scheme is more accurate due to the con-sideration of a list of 64 different types of the nonlinear elastic-plastic waves when constructing the elastic-perfectly plastic Riemann solver.The numerical simulations of typical examples show competitive results.

Extrapolation Cascadic Multigrid Method for Cell-Centered FV Discretization of Diffusion Equations with Strongly Discontinuous and Anisotropic Coefficients

Extrapolation cascadic multigrid(EXCMG)method with conjugate gradient smoother is very efficient for solving the elliptic boundary value problems with linearfinite element discretization.However,it is not trivial to generalize the vertex-centred EXCMG method to cell-centeredfinite volume(FV)methods for diffusion equations with strongly discontinuous and anisotropic coefficients,since a non-nested hierarchy of grid nodes are used in the cell-centered discretization.For cell-centered FV schemes,the vertex values(auxiliary unknowns)need to be approximated by cell-centered ones(primary unknowns).One of the novelties is to propose a new gradient transfer(GT)method of interpolating vertex unknowns with cell-centered ones,which is easy to implement and applicable to general diffusion tensors.The main novelty of this paper is to design a multigrid prolongation operator based on the GT method and splitting extrapolation method,and then propose a cell-centered EXCMG method with BiCGStab smoother for solving the large linear system resulting from linear FV discretization of diffusion equations with strongly discontinuous and anisotropic coefficients.Numerical experiments are presented to demonstrate the high efficiency of the proposed method.

Harten-Lax-van Leer-contact(HLLC)approximation Riemann solver with elastic waves for one-dimensional elastic-plastic problems

A Harten-Lax-van Leer-contact （HLLC） approximate Riemann solver is built with elastic waves （HLLCE） for one-dimensional elastic-plastic flows with a hypo- elastic constitutive model and the von Mises＇ yielding criterion. Based on the HLLCE, a third-order cell-centered Lagrangian scheme is built for one-dimensional elastic-plastic problems. A number of numerical experiments are carried out. The numerical results show that the proposed third-order scheme achieves the desired order of accuracy. The third-order scheme is used to the numerical solution of the problems with elastic shock waves and elastic rarefaction waves. The numerical results are compared with a reference solution and the results obtained by other authors. The comparison shows that the pre- sented high-order scheme is convergent, stable, and essentially non-oscillatory. Moreover, the HLLCE is more efficient than the two-rarefaction Riemann solver with elastic waves （TRRSE）

ADAPTIVE DELAUNAY TRIANGULATION WITH MULTIDIMENSIONAL DISSIPATION SCHEME FOR HIGH-SPEED COMPRESSIBLE FLOW ANALYSIS

Adaptive Delaunay triangulation is combined with the cell-centered upwinding algorithm to analyze inviscid high-speed compressible flow problems. The multidimensional dissipation scheme was developed and included in the upwinding algorithm for unstructured triangular meshes to improve the computed shock wave resolution. The solution accuracy is further improved by coupling an error estimation procedure to a remeshing algorithm that generates small elements in regions with large change of solution gradients, and at the same time, larger elements in other regions. The proposed scheme is further extended to achieve higher-order spatial and temporal solution accuracy. Efficiency of the combined procedure is evaluated by analyzing supersonic shocks and shock propagation behaviors for both the steady and unsteady high-speed compressible flows.

The Confinement Effect of Inert Materials on the Detonation of Insensitive High Explosives

The paper aims to theoretically and numerically investigate the confinement effect of inert materials on the detonation of insensitive high explosives. An improved shock polar theory based on the Zeldovich-von Neumann-Döring model of explosive detonation is established and can fully categorize the confinement interactions between insensitive high explosive and inert materials into six types for the inert materials with smaller sonic velocities than the Chapman-Jouguet velocity of explosive detonation. To confirm the theoretical categorization and obtain the flow details, a second-order, cell-centered Lagrangian hydrodynamic method based on the characteristic theory of the two-dimensional first-order hyperbolic partial differential equations with Ignition-Growth chemistry reaction law is proposed and can exactly numerically simulate the confinement interactions. The numerical result confirms the theoretical categorization and can further merge six types of interaction styles into five types for the inert materials with smaller sonic velocity, moreover, the numerical method can give a new type of interaction style existing a precursor wave in the confining inert material with a larger sonic velocity than the Chapman-Jouguet velocity of explosive detonation, in which a shock polar theory is invalid. The numerical method can also give the effect of inert materials on the edge angles of detonation wave front.

An Unconventional Divergence Preserving Finite-Volume Discretization of Lagrangian Ideal MHD

We construct an unconventional divergence preserving discretization of updated Lagrangian ideal magnetohydrodynamics(MHD)over simplicial grids.The cell-centered finite-volume(FV)method employed to discretize the conservation laws of volume,momentum,and total energy is rigorously the same as the one developed to simulate hyperelasticity equations.By construction this moving mesh method ensures the compatibility between the mesh displacement and the approximation of the volume flux by means of the nodal velocity and the attached unit corner normal vector which is nothing but the partial derivative of the cell volume with respect to the node coordinate under consideration.This is precisely the definition of the compatibility with the Geometrical Conservation Law which is the cornerstone of any proper multi-dimensional moving mesh FV discretization.The momentum and the total energy fluxes are approximated utilizing the partition of cell faces into sub-faces and the concept of sub-face force which is the traction force attached to each sub-face impinging at a node.We observe that the time evolution of the magnetic field might be simply expressed in terms of the deformation gradient which characterizes the Lagrange-to-Euler mapping.In this framework,the divergence of the magnetic field is conserved with respect to time thanks to the Piola formula.Therefore,we solve the fully compatible updated Lagrangian discretization of the deformation gradient tensor for updating in a simple manner the cell-centered value of the magnetic field.Finally,the sub-face traction force is expressed in terms of the nodal velocity to ensure a semi-discrete entropy inequality within each cell.The conservation of momentum and total energy is recovered prescribing the balance of all the sub-face forces attached to the sub-faces impinging at a given node.This balance corresponds to a vectorial system satisfied by the nodal velocity.It always admits a unique solution which provides the nodal velocity.The robustness and the accuracy of this unconventional FV scheme have been demonstrated by employing various representative test cases.Finally,it is worth emphasizing that once you have an updated Lagrangian code for solving hyperelasticity you also get an almost free updated Lagrangian code for solving ideal MHD ensuring exactly the compatibility with the involution constraint for the magnetic field at the discrete level.

A Conservative Lagrangian Scheme for Solving Compressible Fluid Flows with Multiple Internal Energy Equations

Lagrangianmethods arewidely used inmany fields formulti-material compressible flow simulations such as in astrophysics and inertial confinement fusion(ICF),due to their distinguished advantage in capturing material interfaces automatically.In some of these applications,multiple internal energy equations such as those for electron,ion and radiation are involved.In the past decades,several staggeredgrid based Lagrangian schemes have been developed which are designed to solve the internal energy equation directly.These schemes can be easily extended to solve problems with multiple internal energy equations.However such schemes are typically not conservative for the total energy.Recently,significant progress has been made in developing cell-centered Lagrangian schemes which have several good properties such as conservation for all the conserved variables and easiness for remapping.However,these schemes are commonly designed to solve the Euler equations in the form of the total energy,therefore they cannot be directly applied to the solution of either the single internal energy equation or the multiple internal energy equations without significant modifications.Such modifications,if not designed carefully,may lead to the loss of some of the nice properties of the original schemes such as conservation of the total energy.In this paper,we establish an equivalency relationship between the cell-centered discretizations of the Euler equations in the forms of the total energy and of the internal energy.By a carefully designed modification in the implementation,the cell-centered Lagrangian scheme can be used to solve the compressible fluid flow with one or multiple internal energy equations and meanwhile it does not lose its total energy conservation property.An advantage of this approach is that it can be easily applied to many existing large application codes which are based on the framework of solving multiple internal energy equations.Several two dimensional numerical examples for both Euler equations and three-temperature hydrodynamic equations in cylindrical coordinates are presented to demonstrate the performance of the scheme in terms of symmetry preserving,accuracy and non-oscillatory performance.

A Multi-Material CCALE-MOF Approach in Cylindrical Geometry

In this paper we present recent developments concerning a Cell-Centered Arbitrary Lagrangian Eulerian(CCALE)strategy using the Moment Of Fluid(MOF)interface reconstruction for the numerical simulation of multi-material compressible fluid flows on unstructured grids in cylindrical geometries.Especially,our attention is focused here on the following points.First,we propose a new formulation of the scheme used during the Lagrangian phase in the particular case of axisymmetric geometries.Then,the MOF method is considered for multi-interface reconstruction in cylindrical geometry.Subsequently,a method devoted to the rezoning of polar meshes is detailed.Finally,a generalization of the hybrid remapping to cylindrical geometries is presented.These explorations are validated by mean of several test cases using unstructured grid that clearly illustrate the robustness and accuracy of the new method.

Improvement on Spherical Symmetry in Two-Dimensional Cylindrical Coordinates for a Class of Control Volume Lagrangian Schemes

In[14],Maire developed a class of cell-centered Lagrangian schemes for solving Euler equations of compressible gas dynamics in cylindrical coordinates.These schemes use a node-based discretization of the numerical fluxes.The control volume version has several distinguished properties,including the conservation of mass,momentum and total energy and compatibility with the geometric conservation law(GCL).However it also has a limitation in that it cannot preserve spherical symmetry for one-dimensional spherical flow.An alternative is also given to use the first order area-weighted approach which can ensure spherical symmetry,at the price of sacrificing conservation of momentum.In this paper,we apply the methodology proposed in our recent work[8]to the first order control volume scheme of Maire in[14]to obtain the spherical symmetry property.The modified scheme can preserve one-dimensional spherical symmetry in a two-dimensional cylindrical geometry when computed on an equal-angle-zoned initial grid,andmeanwhile itmaintains its original good properties such as conservation and GCL.Several two-dimensional numerical examples in cylindrical coordinates are presented to demonstrate the good performance of the scheme in terms of symmetry,non-oscillation and robustness properties.

Lattice Boltzmann Finite Volume Formulation with Improved Stability

The most severe limitation of the standard Lattice Boltzmann Method is the use of uniform Cartesian grids especiallywhen there is a need for high resolutions near the body or thewalls.Among the recent advances in lattice Boltzmann research to handle complex geometries,a particularly remarkable option is represented by changing the solution procedure from the original"stream and collide"to a finite volume technique.However,most of the presented schemes have stability problems.This paper presents a stable and accurate finite-volume lattice Boltzmann formulation based on a cell-centred scheme.To enhance stability,upwind second order pressure biasing factors are used as flux correctors on a D2Q9 lattice.The resulting model has been tested against a uniform flow past a cylinder and typical free shear flow problems at low and moderate Reynolds numbers:boundary layer,mixing layer and plane jet flows.The numerical results show a very good accuracy and agreement with the exact solution of the Navier-Stokes equation and previous numerical results and/or experimental data.Results in self-similar coordinates are also investigated and show that the timeaveraged statistics for velocity and vorticity express self-similarity at low Reynolds numbers.Furthermore,the scheme is applied to simulate the flow around circular cylinder and the Reynolds number range is chosen in such a way that the flow is time dependent.The agreement of the numerical results with previous results is satisfactory.