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[en] In this paper, 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. Finally, 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.
[en] The edge viscosity of Caramana, Shashkov and Whalen is known to fail on the Noh problem in an initially rectangular grid. In this paper, we present a simple change that significantly improves the behavior in that case. We also show that added energy exchange between cells improves the symmetry of both edge viscosity and the tensor viscosity of Campbell and Shashkov. Finally, as suggested by Noh, this addition also reduces the wall heating effect.
[en] In this paper, we extended the analytical level set method [1, 2] for identifying a piece-wisely heterogeneous (zonation) binary system to the case with an arbitrary number of materials with unknown material properties. In the developed level set approach, starting from an initial guess, the material interfaces are propagated through iterations such that the residuals between the simulated and observed state variables (hydraulic head) is minimized. We derived an expression for the propagation velocity of the interface between any two materials, which is related to the permeability contrast between the materials on two sides of the interface, the sensitivity of the head to permeability, and the head residual. We also formulated an expression for updating the permeability of all materials, which is consistent with the steepest descent of the objective function. The developed approach has been demonstrated through many examples, ranging from totally synthetic cases to a case where the flow conditions are representative of a groundwater contaminant site at the Los Alamos National Laboratory. These examples indicate that the level set method can successfully identify zonation structures, even if the number of materials in the model domain is not exactly known in advance. Although the evolution of the material zonation depends on the initial guess field, inverse modeling runs starting with different initial guesses fields may converge to the similar final zonation structure. These examples also suggest that identifying interfaces of spatially distributed heterogeneities is more important than estimating their permeability values.
[en] Here in this work we combine the dual domain material point method with molecular dynamics in an attempt to create a multiscale numerical method to simulate materials undergoing large deformations with high strain rates. In these types of problems, the material is often in a thermodynamically nonequilibrium state, and conventional constitutive relations or equations of state are often not available. In this method, the closure quantities, such as stress, at each material point are calculated from a molecular dynamics simulation of a group of atoms surrounding the material point. Rather than restricting the multiscale simulation in a small spatial region, such as phase interfaces, or crack tips, this multiscale method can be used to consider nonequilibrium thermodynamic effects in a macroscopic domain. This method takes the advantage that the material points only communicate with mesh nodes, not among themselves; therefore molecular dynamics simulations for material points can be performed independently in parallel. The dual domain material point method is chosen for this multiscale method because it can be used in history dependent problems with large deformation without generating numerical noise as material points move across cells, and also because of its convergence and conservation properties. In conclusion, to demonstrate the feasibility and accuracy of this method, we compare the results of a shock wave propagation in a cerium crystal calculated using the direct molecular dynamics simulation with the results from this combined multiscale calculation.
[en] Least-squares Petrov–Galerkin (LSPG) model-reduction techniques such as the Gauss–Newton with Approximated Tensors (GNAT) method have shown promise, as they have generated stable, accurate solutions for large-scale turbulent, compressible flow problems where standard Galerkin techniques have failed. Furthermore, there has been limited comparative analysis of the two approaches. This is due in part to difficulties arising from the fact that Galerkin techniques perform optimal projection associated with residual minimization at the time-continuous level, while LSPG techniques do so at the time-discrete level. This work provides a detailed theoretical and computational comparison of the two techniques for two common classes of time integrators: linear multistep schemes and Runge–Kutta schemes. We present a number of new findings, including conditions under which the LSPG ROM has a time-continuous representation, conditions under which the two techniques are equivalent, and time-discrete error bounds for the two approaches. Perhaps most surprisingly, we demonstrate both theoretically and computationally that decreasing the time step does not necessarily decrease the error for the LSPG ROM; instead, the time step should be ‘matched’ to the spectral content of the reduced basis. In numerical experiments carried out on a turbulent compressible-flow problem with over one million unknowns, we show that increasing the time step to an intermediate value decreases both the error and the simulation time of the LSPG reduced-order model by an order of magnitude.
[en] Accurate and efficient wave equation modeling is vital for many applications in such as acoustics, electromagnetics, and seismology. However, solving the wave equation in large-scale and highly heterogeneous models is usually computationally expensive because the computational cost is directly proportional to the number of grids in the model. We develop a novel high-order multiscale finite-element method to reduce the computational cost of time-domain acoustic-wave equation numerical modeling by solving the wave equation on a coarse mesh based on the multiscale finite-element theory. In contrast to existing multiscale finite-element methods that use only first-order multiscale basis functions, our new method constructs high-order multiscale basis functions from local elliptic problems which are closely related to the Gauss–Lobatto–Legendre quadrature points in a coarse element. Essentially, these basis functions are not only determined by the order of Legendre polynomials, but also by local medium properties, and therefore can effectively convey the fine-scale information to the coarse-scale solution with high-order accuracy. Numerical tests show that our method can significantly reduce the computation time while maintain high accuracy for wave equation modeling in highly heterogeneous media by solving the corresponding discrete system only on the coarse mesh with the new high-order multiscale basis functions.
[en] Numerical schemes for the determination of stationary axisymmetric toroidal equilibria appropriate for modeling real experimental devices are given. Iterative schemes are used to solve the elliptic nonlinear partial differential equation for the poloidal flux function psi. The principal emphasis is on solving the free boundary (plasma-vacuum interface) equilibrium problem where external current-carrying toroidal coils support the plasma column but fixed boundary (e.g., conducting shell) cases are also included. The toroidal current distribution is given by specifying the pressure and either the poloidal current or the safety factor profiles as functions of psi. Examples of the application of the codes to tokamak design PPPL are given
[en] Here in this work we develop a set of nonlinear correction equations to enforce a consistent time-implicit emission temperature for the original semi-implicit IMC equations. We present two possible forms of correction equations: one results in a set of non-linear, zero-dimensional, non-negative, explicit correction equations, and the other results in a non-linear, non-negative, Boltzman transport correction equation. The zero-dimensional correction equations adheres to the maximum principle for the material temperature, regardless of frequency-dependence, but does not prevent maximum principle violation in the photon intensity, eventually leading to material overheating. The Boltzman transport correction guarantees adherence to the maximum principle for frequency-independent simulations, at the cost of evaluating a reduced source non-linear Boltzman equation. Finally, we present numerical evidence suggesting that the Boltzman transport correction, in its current form, significantly improves time step limitations but does not guarantee adherence to the maximum principle for frequency-dependent simulations.
[en] In this paper, we present a consistent implicit incompressible smoothed particle hydrodynamics (I"2SPH) discretization of Navier–Stokes, Poisson–Boltzmann, and advection–diffusion equations subject to Dirichlet or Robin boundary conditions. It is applied to model various two and three dimensional electrokinetic flows in simple or complex geometries. The accuracy and convergence of the consistent I"2SPH are examined via comparison with analytical solutions, grid-based numerical solutions, or empirical models. Lastly, the new method provides a framework to explore broader applications of SPH in microfluidics and complex fluids with charged objects, such as colloids and biomolecules, in arbitrary complex geometries.
[en] FFT-based solvers introduced in the 1990's for the numerical homogenization of heterogeneous elastic materials have been extended to a wide range of physical properties. In parallel, alternative algorithms and modified discrete Green operators have been proposed to accelerate the method and/or improve the description of the local fields. In this short note, filtering material properties is proposed as a third complementary way to improve FFT-based methods. It is evidenced from numerical experiments that the grid refinement and consequently the computation time and/or the spurious oscillations observed on local fields can be significantly reduced. In addition, if the two filters based on Voigt or Reuss homogenization rules can improve or deteriorate the method depending on the microstructure, a stiff inclusion within a compliant matrix or the reverse, the proposed '2-layer' filter is efficient in both situations. The study is proposed in the context of linear elasticity but similar results are expected in a different physical context (thermal, electrical...). (authors)