Remapping Step Research Articles

Cell-centered indirect Arbitrary Lagrangian-Eulerian numerical strategy for solving 3D gas dynamics equations

Solving the Euler equations under the Lagrangian formalism enables to simulate various complex engineering applications. However, the use of this formalism can lead to significant mesh deformations as the mesh follows the fluid velocity. The mesh quality may be considerably deteriorated requiring a regularization procedure. In the present document, it is shown that the ideas presented by J. Yao (2013)[68] and J. Yao and D. Stillman (2016) [30] may be used efficiently in a 3D hydrodynamics code to perform reliable regularization steps. The flexibility of the methodology in addition to its simplicity, since it only relies on trivial geometrical considerations, opens the way for various extensions. The remapping step then considered, uses the geometrical splitting procedure of the Lagrangian phase to perform effective projections. This coherence ensures the compactness of the overall algorithm. At last but not least, a 3D Flux-Corrected-Remapping method is presented. This yields a particularly robust remapping algorithm while also leading the way for higher order projection extensions.

Collapsing Embedded Cell Complexes for Safer Hexahedral Meshing

We present a set of operators to perform modifications, in particular collapses and splits, in volumetric cell complexes which are discretely embedded in a background mesh. Topological integrity and geometric embedding validity are carefully maintained. We apply these operators strategically to volumetric block decompositions, so-called T-meshes or base complexes, in the context of hexahedral mesh generation. This allows circumventing the expensive and unreliable global volumetric remapping step in the versatile meshing pipeline based on 3D integer-grid maps. In essence, we reduce this step to simpler local cube mapping problems, for which reliable solutions are available. As a consequence, the robustness of the mesh generation process is increased, especially when targeting coarse or block-structured hexahedral meshes. We furthermore extend this pipeline to support feature alignment constraints, and systematically respect these throughout, enabling the generation of meshes that align to points, curves, and surfaces of special interest, whether on the boundary or in the interior of the domain.

Compatible and energy conserving multi-material arbitrary Lagrangian Eulerian scheme for multi-group radiation hydrodynamics simulations

A compatible, bounds preserving and energy conserving multi-material arbitrary Lagrangian Eulerian (ALE) scheme suitable for an unstructured mesh is developed for multi-group radiation hydrodynamics (RHD) simulation of various high energy density physics (HEDP) phenomena involving large material deformations, such as intense thermal radiation or high-power laser driven systems. In order to study very short time scale HEDP phenomena, the electron and ion energy equations (with the effect of electron-ion energy relaxation) are solved separately with an assumption of single macroscopic fluid velocity for each species (two-temperature hydrodynamics model). The governing multi-material hydrodynamics equations are solved by using a higher-order cell-centered hydrodynamics (CCH) scheme. In multi-material flows, the remapping step introduces mixed computational cells in which multiple materials exist in a single cell. Therefore, material interfaces in mixed cells are required to be reconstructed and tracked. In our implementation, this is achieved by using moment-of-fluid (MOF) method. The information of these interfaces is further used to formulate an interface aware sub-scale dynamics (IASSD) scheme for multi-material closure in mixed computational cells. The IASSD scheme is generalized for the two-temperature hydrodynamics model in which separate equations are used to update the electron and ion energies. The multi-group radiation transport on an unstructured mesh is computed using a cell-centered, monotonic, nonlinear finite volume (NLFV) scheme. The flux-limited diffusion (FLD) approximation is used to recover the free-streaming limit of the radiation propagation in optically thin regions. In the ALE step, the electron, ion and radiation energies due to MG-RHD model are conservatively remapped using a center-of-mass (CM) reconstruction of the fields and a compatible and energy conserving remapping scheme. An exact integration based on the intersection between the new (rezoned) and old meshes is used in the remap step. Similar procedure is followed for the remap of cell mass and cell-centered momentum components. In this paper, the details of a compatible, bounds preserving and conservative ALE scheme developed for MG-RHD are described. The remapping of relevant physical quantities due to multi-physics models implemented in the code such as non-local electron energy transport are also discussed. Several validation test problems including laser driven shock propagation in multi-material geometries are provided to demonstrate the accuracy and performance of the ALE scheme implemented.

Arbitrary Lagrangian-Eulerian Remeshing in FE Simulations of Tire Forming

ABSTRACT Tire forming simulation is challenging for Lagrangian finite element method codes due to large changes in the geometry of the tire in the course of molding. This Technical Note briefly describes the use of Arbitrary Lagrangian-Eulerian (ALE) adaptive remeshing in the context of tire molding and curing simulations. The ALE concept generalizes the pure Lagrangian formulation, where the solution within a time step is split into a mesh smoothing step, a history remapping step, and a Lagrangian step. Mesh distortion is reduced in the smoothing step by optimizing the node positions of spatial and material meshes without changing the data structure. The advantage of the ALE approach is demonstrated by a tire forming example.

ALE formulation for thermomechanical inelastic material models applied to tire forming and curing simulations

Forming of tires during production is a challenging process for Lagrangian solid mechanics due to large changes in the geometry and material properties of the rubber layers. This paper extends the Arbitrary Lagrangian–Eulerian (ALE) formulation to thermomechanical inelastic material models with special consideration of rubber. The ALE approach based on tracking the material and spatial meshes is used, and an operator-split is employed which splits up the solution within a time step into a mesh smoothing step, a history remapping step and a Lagrangian step. Mesh distortion is reduced in the smoothing step by solving a boundary value problem. History variables are subsequently remapped to the new mesh with a particle tracking scheme. Within the Lagrangian steps, a fully coupled thermomechanical problem is solved. An advanced two-phase rubber model is incorporated into the ALE approach, which can describe green rubber, cured rubber and the transition process. Several numerical examples demonstrate the superior behavior of the developed formulation in comparison to purely Lagrangian finite elements.

Lagrangian-antidiffusive remap schemes for non-local multi-class traffic flow models

This paper focuses on the numerical approximation of the solutions of a class of non-local systems in one space dimension, arising in traffic modeling. We propose alternative simple schemes by splitting the non-local conservation laws into two different equations, namely the Lagrangian and the remap steps. We provide some properties and estimates recovered by approximating the problem with the Lagrangian-antidiffusive remap (L-AR) scheme, and we prove the convergence to weak solutions in the scalar case. Finally, we show some numerical simulations illustrating the efficiency of the L-AR schemes in comparison with classical first- and second-order numerical schemes.

A random sampling method for a family of Temple-class systems of conservation laws

The Aw–Rascle–Zhang traffic model, a model of sedimentation, and other applications lead to nonlinear $$2 \times 2$$ systems of conservation laws that are governed by a single scalar system velocity. Such systems are of the Temple class since rarefaction wave curves and Hugoniot curves coincide. Moreover, one characteristic field is genuinely nonlinear almost everywhere, and the other is linearly degenerate. Two well-known problems associated with these systems are handled via a random sampling approach. Firstly, Godunov’s and related methods produce spurious oscillations near contact discontinuities since the numerical solution invariably leaves the invariant region of the exact solution. It is shown that alternating between averaging (Av) and remap steps similar to the approach by Chalons and Goatin (Commun Math Sci 5:533–551, 2007) generates numerical solutions that do satisfy an invariant region property. If the remap step is made by random sampling (RS), then combined techniques due to Glimm (Commun Pure Appl Math 18:697–715, 1965), LeVeque and Temple (Trans Am Math Soc 288:115–123, 1985) prove that the resulting Av–RS scheme converges to a weak solution. Numerical examples illustrate that the new scheme is superior to Godunov’s method in accuracy and resolution. Secondly, the vacuum state, which may form even from positive initial data, causes potential problems of non-uniqueness and instability. This is resolved by introducing an alternative Riemann solution concept.

Combined swept region and intersection‐based single‐material remapping method

SummaryA typical arbitrary Lagrangian–Eulerian algorithm consists of a Lagrangian step, where the computational mesh moves with the fluid flow; a rezoning step, where the computational mesh is smoothed and repaired in case it gets too distorted; and a remapping step, where all fluid quantities are conservatively interpolated on this new mesh. In single‐material simulations, the remapping process can be represented in a flux form, with fluxes approximated by integrating a reconstructed function over intersections of neighboring computational cells on the original and rezoned computational mesh. This algorithm is complex and computationally demanding – Therefore, a simpler approach that utilizes regions swept by the cell edges during rezoning is often used in practice. However, it has been observed that such simplification can lead to distortion of the solution symmetry, especially when the mesh movement is not bound by such symmetry.For this reason, we propose an algorithm combining both approaches in a similar way that was proposed for multi‐material remapping (two‐step hybrid remap). Intersections and exact integration are employed only in certain parts of the computational mesh, marked by a switching function – Various different criteria are presented in this paper. The swept‐based method is used elsewhere in areas that are not marked. This way, our algorithm can retain the beneficial symmetry‐preserving capabilities of intersection‐based remapping while keeping the overall computational cost moderate.

A 4th-Order Particle-in-Cell Method with Phase-Space Remapping for the Vlasov--Poisson Equation

Numerical solutions to the Vlasov--Poisson system of equations have important applications to both plasma physics and cosmology. In this paper, we present a new particle-in-cell (PIC) method for solving this system that is 4th-order accurate in both space and time. Our method is a high-order extension of one presented previously [B. Wang, G. H. Miller, and P. Colella, SIAM J. Sci. Comput., 33 (2011), pp. 3509--3537]. It treats all of the stages of the standard PIC update---charge deposition, force interpolation, the field solve, and the particle push---with 4th-order accuracy, and includes a 6th-order accurate phase-space remapping step for controlling particle noise. We demonstrate the convergence of our method on a series of one- and two- dimensional electrostatic plasma test problems, comparing its accuracy to that of a 2nd-order method. As expected, the 4th-order method can achieve accuracy comparable to that of the 2nd-order method and with considerably fewer resolution elements.

Antidiffusive Lagrangian‐remap schemes for models of polydisperse sedimentation

One‐dimensional models of gravity‐driven sedimentation of polydisperse suspensions with particles that belong to N size classes give rise to systems of N strongly coupled, nonlinear first‐order conservation laws for the local solids volume fractions. As the eigenvalues and eigenvectors of the flux Jacobian have no closed algebraic form, characteristic‐wise numerical schemes for these models become involved. Alternative simple schemes for this model directly utilize the velocity functions and are based on splitting the system of conservation laws into two different first‐order quasi‐linear systems, which are solved successively for each time iteration, namely, the Lagrangian and remap steps (so‐called Lagrangian‐remap [LR] schemes). This approach was advanced in (Bürger, Chalons, and Villada, SIAM J Sci Comput 35 (2013), B1341–B1368) for a multiclass Lighthill–Whitham‐Richards traffic model with nonnegative velocities. By incorporating recent antidiffusive techniques for transport equations a new version of these Lagrangian‐antidiffusive remap (L‐AR) schemes for the polydisperse sedimentation model is constructed. These L‐AR schemes are supported by a partial analysis for N = 1. They are total variation diminishing under a suitable CFL condition and therefore converge to a weak solution. Numerical examples illustrate that these schemes, including a more accurate version based on MUSCL extrapolation, are competitive in accuracy and efficiency with several existing schemes. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1109–1136, 2016

Numerical simulations of instabilities in the implosion process of inertial confined fusion in 2D cylindrical coordinates

In this paper, we introduce a multi-material arbitrary Lagrangian and Eulerian method for the hydrodynamic radiative multi-group diffusion model in 2D cylindrical coordinates. The basic idea in the construction of the method is the following: In the Lagrangian step, a closure model of radiation-hydrodynamics is used to give the states of equations for materials in mixed cells. In the mesh rezoning step, we couple the rezoning principle with the Lagrangian interface tracking method and an Eulerian interface capturing scheme to compute interfaces sharply according to their deformation and to keep cells in good geometric quality. In the interface reconstruction step, a dual-material Moment-of-Fluid method is introduced to obtain the unique interface in mixed cells. In the remapping step, a conservative remapping algorithm of conserved quantities is presented. A number of numerical tests are carried out and the numerical results show that the new method can simulate instabilities in complex fluid field under large deformation, and are accurate and robust.

A swept intersection‐based remapping method for axisymmetric ReALE computation

SummaryWe use here a reconnection ALE (ReALE) strategy to solve hydrodynamic compressible flows in cylindrical geometries. The main difference between the classical ALE and the ReALE method is the rezoning step where we allow change in the topology. This leads for ReALE to a polygonal mesh, which follows more efficiently the flow. We present here a new displacement of generators in order to keep the Lagrangian features, which are usually lost using ALE with fixed topology. The reconnection capability allows to deal with complex geometries and high‐vorticity problems contrary to ALE method. The main difficulty of ReALE is the remapping step where we have to remap physical variables on a mesh with a different topology. For this step, a new remapping method based on a swept intersection algorithm has been developed in the case of planar geometries. We present here the extension of the swept intersection‐based remapping method to cylindrical geometries. We demonstrate that our method can be applied to several numerical examples up to problem representative of hydrodynamic experiments. Copyright © 2015 John Wiley & Sons, Ltd.

Comparison of different numerical approaches to the 1D sea-ice thermodynamics problem

The vertical one-dimensional sea-ice thermodynamic problem using the principle of conservation of enthalpy is revisited here using (1) the Bitz and Lipscomb (1999) finite-difference approach (FD), (2) a reformulation of the sigma-level transformation of Huwald et al. (2005b) (FV) and (3) a Finite Element approach also in sigma coordinates (FE). These three formulations are compared in terms of physics, numerics, and performance, in order to identify the best choice for large-scale climate models. The BL99 formulation sequentially treats the diffusion of heat and the changes in the vertical position of the ice-snow layers. In contrast, the FV sigma-level transformation elegantly treats both simultaneously. The original FV formulation suffers however from slow convergence. The convergence can nonetheless be improved significantly with a few simple modifications to the original code. The three formulations are compared following the experimental protocol of the Sea Ice Model Intercomparison Project for ice thermodynamics (SIMIP2). It is found that all formulations converge to the same solution. The FD approach, however, suffers from the added cost of the remapping step at large number of ice layers (we include in the appendix an optimized version of the FD code–written by one of the reviewer–that resolves this issue). Finally the FE formulation results in a sub-surface temperature over-estimation at low resolution, a problem which disappears at high resolution. Hence, only FD and FV are found suitable for climate models.

A frame invariant and maximum principle enforcing second‐order extension for cell‐centered ALE schemes based on local convex hull preservation

SUMMARYTwo difficulties are clearly identified for high‐order extensions of ALE schemes for Euler equations: strict respect of the maximum principle and preservation of the Galilean invariance. We deal with these two issues in this paper. Our approach is closely related to the concepts of a posteriori limiting and convex hull spanning. We introduce the notion of local convex hull preservation schemes, which embodies these two concepts. We lean on this notion to propose a fully Galilean invariant ALE scheme. Moreover, we provide a new limiter (called Apitali for A Posteriori ITerAtive LImiter) for the remap step, enforcing the local convex hull preservation property. Copyright © 2014 John Wiley & Sons, Ltd.

Numerical study on the water impact of 3D bodies by an explicit finite element method

The hydrodynamic problem of the water impact of three-dimensional buoys is investigated by the explicit finite element method with an Arbitrary-Lagrangian Eulerian (ALE) solver. The fluid is solved by using an Eulerian formulation, while the structure is discretized by a Lagrangian approach, and a penalty coupling algorithm enables the interaction between the body and the fluids. The remap step in the ALE algorithm applies a donor cell+HIS (Half-Index-Shift) advection algorithm to update fluid velocity and history variables. The interface between the solid structure and the fluids is captured by Volume of Fluid method. Convergence studies are carried out for three dimensional hemisphere and cones with different deadrise angles. It is found that the mesh density of the impact domain is very important to the quality of the simulation results. The contact stiffness between the coupling nodes affects the local peak pressure values. The numerical calculations are validated by comparing with other available results, for both the drop cases and the ones with constant impact velocity.

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.

Antidiffusive and Random-Sampling Lagrangian-Remap Schemes for the Multiclass Lighthill--Whitham--Richards Traffic Model

The multiclass Lighthill--Whitham--Richards (MCLWR) traffic model, which distinguishes $N$ classes of drivers differing in preferential velocity, gives rise to a system of $N$ strongly coupled, nonlinear first-order conservation laws for the car densities as a function of distance and time. The corresponding velocities involve a hindrance function that depends on the local total density of cars. Since the eigenvalues and eigenvectors of the flux Jacobian have no closed algebraic form, characteristic-wise numerical schemes for the MCLWR model become involved. Alternative simple schemes for this model directly utilize that the velocity functions are nonnegative and strictly decreasing, which allows one to construct a new class of schemes by splitting the system of conservation laws into two different first-order quasi-linear systems, which are solved successively for each time iteration, namely, the Lagrangian and remap steps. The new schemes are called Lagrangian-remap (LR) schemes. One version of LR schemes incorporates recent antidiffusive techniques for transport equations. The corresponding subclass of LR schemes are called Lagrangian-antidiffusive-remap (L-AR) schemes. Alternatively, the remap step can be handled by Glimm-like random sampling, which gives rise to a statistically conservative Lagrangian-random-sampling (L-RS) scheme that is less diffusive than other remap techniques. The LR schemes for the MCLWR model are supported by a partial analysis of the L-AR schemes for $N=1$, which are total variation diminishing under a suitable CFL condition and therefore converge to a weak solution, and by numerical examples for both L-AR and L-RS subclasses of schemes.

Adaptive methods for multi‐material ALE hydrodynamics

AbstractArbitrary Lagrangian–Eulerian (ALE) methods are commonly used for challenging problems in hydrodynamics. Among the most challenging matters are the approximations in the presence of multiple materials. The ALE code ALEGRA has used a constant volume method for computing the impact of multiple materials on both the Lagrangian step and the remap step of the method. Here, we describe modifications to these methods that provide greater modeling fidelity and better numerical and computational performance. In the Lagrangian step, the effects of differences in material response were not included in the constant volume method, but have been included in the new method. The new methodology can produce unstable results unless the changes in the variable states are carefully controlled. Both the stability analysis and the control of the instability are described. In the standard (Van Leer) method for the remap, the numerical approximation did not account for the presence of a material interface directly. The new methodology uses different, more stable and more dissipative numerical approximations in and near material interfaces. In addition, the standard numerical method, which is second‐order accurate, has been replaced by a more accurate method, which is third‐order accurate in one dimension. Published in 2010 by John Wiley & Sons, Ltd.

High-order regridding–remapping schemes for continuous isopycnal and generalized coordinates in ocean models

A hierarchy of high-order regridding–remapping schemes for use in generalized vertical coordinate ocean models is presented. The proposed regridding–remapping framework is successfully used in a series of idealized one-dimensional numerical experiments as well as two-dimensional internal wave and overflow test cases. The model is capable of replicating z-, sigma- and isopycnal-coordinate results, among others. Particular emphasis is placed on the design of a continuous isopycnal framework, which is a more general alternative to the layered isopycnal paradigm. Continuous isopycnal coordinates use target interface densities to define layers. In contrast to traditional layered isopycnal models, in which along-layer density gradients vanish, general coordinate approaches must deal with extra terms. For example, the calculation of pressure gradient force is more complicated and must be evaluated carefully. High-order reconstructions within boundary cells are crucial for obtaining sensible results and for reducing spurious diffusion near boundaries. Vertical advection is implicitly embedded in the remapping step and directly benefits from high-order schemes. Volume and all tracers are conserved to machine precision, which is a necessary ingredient for long-term ocean climate modeling. This hybrid vertical coordinate model provides the framework to easily capture the impact of different coordinate systems on dynamics.

Remapping-free ALE-type kinetic method for flow computations

Based on the integral form of the fluid dynamic equations, a finite volume kinetic scheme with arbitrary control volume and mesh velocity is developed. Different from the earlier unified moving mesh gas-kinetic method [C.Q. Jin, K. Xu, An unified moving grid gas-kinetic method in Eulerian space for viscous flow computation, J. Comput. Phys. 222 (2007) 155–175], the coupling of the fluid equations and geometrical conservation laws has been removed in order to make the scheme applicable for any quadrilateral or unstructured mesh rather than parallelogram in 2D case. Since a purely Lagrangian method is always associated with mesh entangling, in order to avoid computational collapsing in multidimensional flow simulation, the mesh velocity is constructed by considering both fluid velocity (Lagrangian methodology) and diffusive velocity (Regenerating Eulerian mesh function). Therefore, we obtain a generalized Arbitrary-Lagrangian–Eulerian (ALE) method by properly designing a mesh velocity instead of re-generating a new mesh after distortion. As a result, the remapping step to interpolate flow variables from old mesh to new mesh is avoided. The current method provides a general framework, which can be considered as a remapping-free ALE-type method. Since there is great freedom in choosing mesh velocity, in order to improve the accuracy and robustness of the method, the adaptive moving mesh method [H.Z. Tang, T. Tang, Adaptive mesh methods for one-and two-dimensional hyperbolic conservation laws, SIAM J. Numer. Anal. 41 (2003) 487–515] can be also used to construct a mesh velocity to concentrate mesh to regions with high flow gradients.