Donald E. Burton

An approach for treating contact surfaces in Lagrangian cell-centered hydrodynamics

A new method is presented for modeling contact surfaces in Lagrangian cell-centered hydrodynamics (CCH). The contact method solves a multi-directional Riemann-like problem at each penetrating or touching node along the contact surface. The velocity of a penetrating or touching node and the corresponding forces are explicitly calculated using the Riemann-like nodal solver. The contact method works with material strength and allows surfaces to impact, slide, and separate. Results are presented for several test problems involving both gases and materials with strength. The new contact surface approach extends the modeling capabilities of CCH.

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.

Reduction of dissipation in Lagrange cell-centered hydrodynamics (CCH) through corner gradient reconstruction (CGR)

This work presents an extension of a second order cell-centered hydrodynamics scheme on unstructured polyhedral cells 13 toward higher order. The goal is to reduce dissipation, especially for smooth flows. This is accomplished by multiple piecewise linear reconstructions of conserved quantities within the cell. The reconstruction is based upon gradients that are calculated at the nodes, a procedure that avoids the least-square solution of a large equation set for polynomial coefficients. Conservation and monotonicity are guaranteed by adjusting the gradients within each cell corner. Results are presented for a wide variety of test problems involving smooth and shock-dominated flows, fluids and solids, 2D and 3D configurations, as well as Lagrange, Eulerian, and ALE methods.

A Godunov-like point-centered essentially Lagrangian hydrodynamic approach

We present an essentially Lagrangian hydrodynamic scheme suitable for modeling complex compressible flows on tetrahedron meshes. The scheme reduces to a purely Lagrangian approach when the flow is linear or if the mesh size is equal to zero; as a result, we use the term essentially Lagrangian for the proposed approach. The motivation for developing a hydrodynamic method for tetrahedron meshes is because tetrahedron meshes have some advantages over other mesh topologies. Notable advantages include reduced complexity in generating conformal meshes, reduced complexity in mesh reconnection, and preserving tetrahedron cells with automatic mesh refinement. A challenge, however, is tetrahedron meshes do not correctly deform with a lower order (i.e. piecewise constant) staggered-grid hydrodynamic scheme (SGH) or with a cell-centered hydrodynamic (CCH) scheme. The SGH and CCH approaches calculate the strain via the tetrahedron, which can cause artificial stiffness on large deformation problems. To resolve the stiffness problem, we adopt the point-centered hydrodynamic approach (PCH) and calculate the evolution of the flow via an integration path around the node. The PCH approach stores the conserved variables (mass, momentum, and total energy) at the node. The evolution equations for momentum and total energy are discretized using an edge-based finite element (FE) approach with linear basis functions. A multidirectional Riemann-like problem is introduced at the center of the tetrahedron to account for discontinuities in the flow such as a shock. Conservation is enforced at each tetrahedron center. The multidimensional Riemann-like problem used here is based on Lagrangian CCH work 8,19,37,38,44 and recent Lagrangian SGH work 33-35,39,45. In addition, an approximate 1D Riemann problem is solved on each face of the nodal control volume to advect mass, momentum, and total energy. The 1D Riemann problem produces fluxes 18 that remove a volume error in the PCH discretization. A 2-stage Runge-Kutta method is used to evolve the solution in time. The details of the new hydrodynamic scheme are discussed; likewise, results from numerical test problems are presented.

A point-centered arbitrary Lagrangian Eulerian hydrodynamic approach for tetrahedral meshes

We present a three dimensional (3D) arbitrary Lagrangian Eulerian (ALE) hydrodynamic scheme suitable for modeling complex compressible flows on tetrahedral meshes. The new approach stores the conserved variables (mass, momentum, and total energy) at the nodes of the mesh and solves the conservation equations on a control volume surrounding the point. This type of an approach is termed a point-centered hydrodynamic (PCH) method. The conservation equations are discretized using an edge-based finite element (FE) approach with linear basis functions. All fluxes in the new approach are calculated at the center of each tetrahedron. A multidirectional Riemann-like problem is solved at the center of the tetrahedron. The advective fluxes are calculated by solving a 1D Riemann problem on each face of the nodal control volume. A 2-stage Runge-Kutta method is used to evolve the solution forward in time, where the advective fluxes are part of the temporal integration. The mesh velocity is smoothed by solving a Laplacian equation. The details of the new ALE hydrodynamic scheme are discussed. Results from a range of numerical test problems are presented.

Compatible, energy and symmetry preserving 2D lagrangian hydrodynamics in rz — cylindrical coordinates

We present a new discretization for 2D Lagrangian hydrodynamics in rz geometry (cylindrical coordinates), which is total energy conserving and symmetry preserving.

Compatible, energy conserving, bounds preserving remap of hydrodynamic fields for an extended ALE scheme

Abstract From the very origins of numerical hydrodynamics in the Lagrangian work of von Neumann and Richtmyer [83] , the issue of total energy conservation as well as entropy production has been problematic. Because of well known problems with mesh deformation, Lagrangian schemes have evolved into Arbitrary Lagrangian–Eulerian (ALE) methods [39] that combine the best properties of Lagrangian and Eulerian methods. Energy issues have persisted for this class of methods. We believe that fundamental issues of energy conservation and entropy production in ALE require further examination. The context of the paper is an ALE scheme that is extended in the sense that it permits cyclic or periodic remap of data between grids of the same or differing connectivity. The principal design goals for a remap method then consist of total energy conservation, bounded internal energy, and compatibility of kinetic energy and momentum. We also have secondary objectives of limiting velocity and stress in a non-directional manner, keeping primitive variables monotone, and providing a higher than second order reconstruction of remapped variables. In particular, the new contributions fall into three categories associated with: energy conservation and entropy production, reconstruction and bounds preservation of scalar and tensor fields, and conservative remap of nonlinear fields. The paper presents a derivation of the methods, details of implementation, and numerical results for a number of test problems. The methods requires volume integration of polynomial functions in polytopal cells with planar facets, and the requisite expressions are derived for arbitrary order.

Mimetic Theory for Cell-Centered Lagrangian Finite Volume Formulation on General Unstructured Grids

A finite volume cell-centered Lagrangian scheme for solving large deformation problems is constructed based on the hypo-elastic model and using the mimetic theory. Rigorous analysis in the context of gas and solid dynamics, and arbitrary polygonal meshes, is presented to demonstrate the ability of cell-centered schemes in mimicking the continuum properties and principles at the discrete level. A new mimetic formulation based gradient evaluation technique and physics-based, frame independent and symmetry preserving slope limiters are proposed. Furthermore, a physically consistent dissipation model is employed which is both robust and inexpensive to implement. The cell-centered scheme along with these additional new features are applied to solve solids undergoing elasto-plastic deformation.

Reducing spurious mesh motion in Lagrangian finite volume and discontinuous Galerkin hydrodynamic methods

Abstract The Lagrangian finite volume (FV) cell-centered hydrodynamic (CCH) method and the Lagrangian discontinuous Galerkin (DG) CCH method have been demonstrated to be quite stable and capable of producing very accurate solutions on many mesh topologies. However, some challenges can arise with higher-order elements and polygonal elements that have many deformational degrees of freedom. With these types of meshes, elements can deform in unphysical ways and the mesh can tangle. We present methods for obtaining more robust Lagrangian solutions on polygonal and higher-order elements. The robustness is achieved by (1) incorporating a new iterative method that modifies the velocity reconstructions in the corners of the elements, and (2) a new multidirectional approximate Riemann solver that, when coupled with the iterative method, reduces spurious mesh motion. The details of the numerical methods are discussed and their utility is demonstrated on a diverse suite of test problems using higher-order and polygonal elements.

Lagrangian discontinuous Galerkin hydrodynamic methods in axisymmetric coordinates

Abstract We present new Lagrangian discontinuous Galerkin (DG) hydrodynamic methods for compressible flows on unstructured meshes in axisymmetric coordinates. The physical evolution equations for the specific volume, velocity, and specific total energy are discretized using a modal DG method with linear Taylor series polynomials. Two different approaches are used to discretize the evolution equations – the first one is the true volume approach and the second one is the area-weighted approach. For the true volume approach, the DG equations are derived using the true 3D volume that is consistent with the geometry conservation law (GCL). The Riemann velocity at the nodes on the surface of the element, and the corresponding surface forces, are calculated by solving a multidirectional approximate Riemann problem using surfaces areas for axisymmetric coordinates. This true volume approach conserves mass, momentum, and total energy and satisfies the GCL. However, it can not preserve spherical symmetry on an equal-angle polar grid with 1D radial flows. For the area-weighted approach, the DG equations are based on the 2D Cartesian geometry that is rotated about the axis of symmetry using a single, element average radius. With this approach, the Riemann velocity at the nodes on the surface of the element, and the corresponding surface forces, are calculated by solving a multidirectional approximate Riemann problem in 2D Cartesian geometry. This area-weighted approach, in the limit of an infinitesimal mesh size, conserves physical momentum, and physical total energy. The area-weighted approach preserves spherical symmetry on an equal-angle polar grid for 1D radial flows, but it does not satisfy the GCL. A suite of test problems are calculated to demonstrate stable mesh motion, the expected second order accuracy of these methods, and that the new area-weighted DG method preserves spherical symmetry on 1D radial flow problems with equal-angle polar meshes.