Pressure boundary conditions for computing incompressible flows with SPH

In Smoothed Particle Hydrodynamics (SPH) methods for fluid flow, incompressibility may be imposed by a projection method with an artificial homogeneous Neumann boundary condition for the pressure Poisson equation. This is often inconsistent with physical conditions at solid walls and inflow and outflow boundaries. For this reason open-boundary flows have rarely been computed using SPH. In this work, we demonstrate that the artificial pressure boundary condition produces a numerical boundary layer that compromises the solution near boundaries. We resolve this problem by utilizing a “rotational pressure-correction scheme” with a consistent pressure boundary condition that relates the normal pressure gradient to the local vorticity. We show that this scheme computes the pressure and velocity accurately near open boundaries and solid objects, and extends the scope of SPH simulation beyond the usual periodic boundary conditions.     

An MPI parallel level-set algorithm for propagating front curvature dependent detonation shock fronts in complex geometries

We present a parallel, two-dimensional, grid-based algorithm for solving a level-set function PDE that arises in Detonation Shock Dynamics (DSD). In the DSD limit, the detonation shock propagates at a speed that is a function of the curvature of the shock surface, subject to a set of boundary conditions applied along the boundaries of the detonating explosive. Our method solves for the full level-set function field, φ(x, y, t), that locates the detonation shock with a modified level-set function PDE that continuously renormalises the level-set function to a distance function based off of the locus of the shock surface, φ(x, y, t)=0. The boundary conditions are applied with ghost nodes that are sorted according to their connectivity to the interior explosive nodes. This allows the boundary conditions to be applied via a local, direct evaluation procedure. We give an extension of this boundary condition application method to three dimensions. Our parallel algorithm is based on a domain-decomposition model which uses the Message-Passing Interface (MPI) paradigm. The computational order of the full level-set algorithm, which is O(N ⁴), where N is the number of grid points along a coordinate line, makes an MPI-based algorithm an attractive alternative. This parallel model partitions the overall explosive domain into smaller sub-domains which in turn get mapped onto processors that are topologically arranged into a two-dimensional rectangular grid. A comparison of our numerical solution with an exact solution to the problem of a detonation rate stick shows that our numerical solution converges at better than first-order accuracy as measured by an L1-norm. This represents an improvement over the convergence properties of narrow-band level-set function solvers, whose convergence is limited to a floor set by the width of the narrow band. The efficiency of the narrow-band method is recovered by using our parallel model.     

Efficient implementation of high order inverse Lax–Wendroff boundary treatment for conservation laws

In [20], two of the authors developed a high order accurate numerical boundary condition procedure for hyperbolic conservation laws, which allows the computation using high order finite difference schemes on Cartesian meshes to solve problems in arbitrary physical domains whose boundaries do not coincide with grid lines. This procedure is based on the so-called inverse Lax–Wendroff (ILW) procedure for inflow boundary conditions and high order extrapolation for outflow boundary conditions. However, the algebra of the ILW procedure is quite heavy for two dimensional (2D) hyperbolic systems, which makes it difficult to implement the procedure for order of accuracy higher than three. In this paper, we first discuss a simplified and improved implementation for this procedure, which uses the relatively complicated ILW procedure only for the evaluation of the first order normal derivatives. Fifth order WENO type extrapolation is used for all other derivatives, regardless of the direction of the local characteristics and the smoothness of the solution. This makes the implementation of a fifth order boundary treatment practical for 2D systems with source terms. For no-penetration boundary condition of compressible inviscid flows, a further simplification is discussed, in which the evaluation of the tangential derivatives involved in the ILW procedure is avoided. We test our simplified and improved boundary treatment for Euler equations with or without source terms representing chemical reactions in detonations. The results demonstrate the designed fifth order accuracy, stability, and good performance for problems involving complicated interactions between detonation/shock waves and solid boundaries.     

Universal quadratures for boundary integral equations on two-dimensional domains with corners

We describe the construction of a collection of quadrature formulae suitable for the efficient discretization of certain boundary integral equations on a very general class of two-dimensional domains with corner points. The resulting quadrature rules allow for the rapid high-accuracy solution of Dirichlet boundary value problems for Laplace’s equation and the Helmholtz equation on such domains under a mild assumption on the boundary data. Our approach can be adapted to other boundary value problems and certain aspects of our scheme generalize to the case of surfaces with singularities in three dimensions. The performance of the quadrature rules is illustrated with several numerical examples.     

On the use of immersed boundary methods for shock/obstacle interactions

This paper describes the implementation of immersed boundary method using the direct-forcing concept to investigate complex shock–obstacle interactions. An interpolation algorithm is developed for more stable boundary conditions with easier implementation procedure. The values of the fluid variables at the embedded ghost-cells are obtained using a local quadratic scheme which involves the neighboring fluid nodes. Detailed discussions of the method are presented on the interpolation of flow variables, direct-forcing of ghost cells, resolution of immersed-boundary points and internal treatment. The method is then applied to a high-order WENO scheme to simulate the complex fluid–solid interactions. The developed solver is first validated against the theoretical solutions of supersonic flow past triangular prism and circular cylinder. Simulated results for test cases with moving shocks are further compared with the previous experimental results of literature in terms of triple-point trajectory and vortex evolution. Excellent agreement is obtained showing the accuracy and the capability of the proposed method for solving complex strong-shock/obstacle interactions for both stationary and moving shock waves.     

Implementation of nonreflecting boundary conditions for the nonlinear Euler equations

Computationally efficient nonreflecting boundary conditions are derived for the Euler equations with acoustic, entropic and vortical inflow disturbances. The formulation linearizes the Euler equations near the inlet/outlet boundaries and expands the solution in terms of Fourier–Bessel modes. This leads to an ‘exact’ nonreflecting boundary condition, local in space but nonlocal in time, for each Fourier–Bessel mode of the perturbation pressure. The perturbation velocity and density are then calculated using acoustic, entropic and vortical mode splitting. Extension of the boundary conditions to nonuniform swirling flows is presented for the narrow annulus limit which is relevant to many aeroacoustic problems. The boundary conditions are implemented for the nonlinear Euler equations which are solved in space using the finite volume approximation and integrated in time using a MacCormack scheme. Two test problems are carried out: propagation of acoustic waves in an annular duct and the scattering of a vortical wave by a cascade. Comparison between the present exact conditions and commonly used approximate local boundary conditions is made. Results show that, unlike the local boundary conditions whose accuracy depends on the group velocity of the scattered waves, the present conditions give accurate solutions for a range of problems that have a wide array of group velocities. Results also show that this approach leads to a significant savings in computational time and memory by obviating the need to store the pressure field and calculate the nonlocal convolution integral at each point in the inlet and exit boundaries.     

A novel mesh regeneration algorithm for 2D FEM simulations of flows with moving boundary

A novel mesh regeneration algorithm is proposed to maintain the mesh structure during a finite element simulation of flows with moving solid boundary. With the current algorithm, a new body-fitted mesh can be efficiently constructed by solving a set of Laplace equations developed to specify the displacements of individual mesh elements. These equations are subjected to specific boundary conditions determined by the instantaneous body motion and other flow boundary conditions. The proposed mesh regeneration algorithm has been implemented on an arbitrary Lagrangian–Eulerian (ALE) framework that employs an operator-splitting technique to solve the Navier–Stokes equations. The integrated numerical scheme was validated by the numerical results of four existing problems: a flow over a backward-facing step, a uniform flow over a fixed cylinder, the vortex-induced vibration of an elastic cylinder in uniformly incident flow, and a complementary problem that compares the transient drag coefficient for a cylinder impulsively set into motion to that measured on a fixed cylinder in a starting flow. Good agreement with the numerical or experimental data in the literature was obtained and new transient flow dynamics was revealed. The scheme performance is further examined with respect to the parameter employed in the mesh regeneration algorithm.     

A high-resolution mapped grid algorithm for compressible multiphase flow problems

We describe a simple mapped-grid approach for the efficient numerical simulation of compressible multiphase flow in general multi-dimensional geometries. The algorithm uses a curvilinear coordinate formulation of the equations that is derived for the Euler equations with the stiffened gas equation of state to ensure the correct fluid mixing when approximating the equations numerically with material interfaces. A γ-based and a α-based model have been described that is an easy extension of the Cartesian coordinates counterpart devised previously by the author [30]. A standard high-resolution mapped grid method in wave-propagation form is employed to solve the proposed multiphase models, giving the natural generalization of the previous one from single-phase to multiphase flow problems. We validate our algorithm by performing numerical tests in two and three dimensions that show second order accurate results for smooth flow problems and also free of spurious oscillations in the pressure for problems with interfaces. This includes also some tests where our quadrilateral-grid results in two dimensions are in direct comparisons with those obtained using a wave-propagation based Cartesian grid embedded boundary method.     

A simple multigrid scheme for solving the Poisson equation with arbitrary domain boundaries

We present a new multigrid scheme for solving the Poisson equation with Dirichlet boundary conditions on a Cartesian grid with irregular domain boundaries. This scheme was developed in the context of the Adaptive Mesh Refinement (AMR) schemes based on a graded-octree data structure. The Poisson equation is solved on a level-by-level basis, using a “one-way interface” scheme in which boundary conditions are interpolated from the previous coarser level solution. Such a scheme is particularly well suited for self-gravitating astrophysical flows requiring an adaptive time stepping strategy. By constructing a multigrid hierarchy covering the active cells of each AMR level, we have designed a memory-efficient algorithm that can benefit fully from the multigrid acceleration. We present a simple method for capturing the boundary conditions across the multigrid hierarchy, based on a second-order accurate reconstruction of the boundaries of the multigrid levels. In case of very complex boundaries, small scale features become smaller than the discretization cell size of coarse multigrid levels and convergence problems arise. We propose a simple solution to address these issues. Using our scheme, the convergence rate usually depends on the grid size for complex grids, but good linear convergence is maintained. The proposed method was successfully implemented on distributed memory architectures in the RAMSES code, for which we present and discuss convergence and accuracy properties as well as timing performances.     

A simple second order cartesian scheme for compressible Euler flows

We present a finite-volume scheme for compressible Euler flows where the grid is cartesian and it does not fit to the body. The scheme, based on the definition of an ad hoc Riemann problem at solid boundaries, is simple to implement and it is formally second order accurate. Error convergence rates with respect to several exact test cases are investigated and examples of flow solutions in one, two and three dimensions are presented.     

A differentially interpolated direct forcing immersed boundary method for predicting incompressible Navier–Stokes equations in time-varying complex geometries

A dispersion-relation-preserving dual-compact scheme developed in Cartesian grids is applied together with the immersed boundary method to solve the flow equations in irregular and time-varying domains. The artificial momentum forcing term applied at certain points in cells containing fluid and solid allows an imposition of velocity condition to account for the motion of solid body. We develop in this study a differential-based interpolation scheme which can be easily extended to three-dimensional simulation. The results simulated from the proposed immersed boundary method agree well with other numerical and experimental results for the chosen benchmark problems. The accuracy and fidelity of the IB flow solver developed to predict flows with irregular boundaries are therefore demonstrated.     

Inverse Lax-Wendroff procedure for numerical boundary conditions of conservation laws

We develop a high order finite difference numerical boundary condition for solving hyperbolic conservation laws on a Cartesian mesh. The challenge results from the wide stencil of the interior high order scheme and the fact that the boundary intersects the grids in an arbitrary fashion. Our method is based on an inverse Lax-Wendroff procedure for the inflow boundary conditions. We repeatedly use the partial differential equation to write the normal derivatives to the inflow boundary in terms of the time derivatives and the tangential derivatives. With these normal derivatives, we can then impose accurate values of ghost points near the boundary by a Taylor expansion. At outflow boundaries, we use Lagrange extrapolation or least squares extrapolation if the solution is smooth, or a weighted essentially non-oscillatory (WENO) type extrapolation if a shock is close to the boundary. Extensive numerical examples are provided to illustrate that our method is high order accurate and has good performance when applied to one and two-dimensional scalar or system cases with the physical boundary not aligned with the grids and with various boundary conditions including the solid wall boundary condition. Additional numerical cost due to our boundary treatment is discussed in some of the examples.     

A high order moving boundary treatment for compressible inviscid flows

We develop a high order numerical boundary condition for compressible inviscid flows involving complex moving geometries. It is based on finite difference methods on fixed Cartesian meshes which pose a challenge that the moving boundaries intersect the grid lines in an arbitrary fashion. Our method is an extension of the so-called inverse Lax–Wendroff procedure proposed in [17] for conservation laws in static geometries. This procedure helps us obtain normal spatial derivatives at inflow boundaries from Lagrangian time derivatives and tangential derivatives by repeated use of the Euler equations. Together with high order extrapolation at outflow boundaries, we can impose accurate values of ghost points near the boundaries by a Taylor expansion. To maintain high order accuracy in time, we need some special time matching technique at the two intermediate Runge–Kutta stages. Numerical examples in one and two dimensions show that our boundary treatment is high order accurate for problems with smooth solutions. Our method also performs well for problems involving interactions between shocks and moving rigid bodies.     

Schwarz method for earthquake source dynamics

Dynamic faulting under slip-dependent friction in a linear elastic domain (in-plane and 3D configurations) is considered. The use of an implicit time-stepping scheme (Newmark method) allows much larger values of the time step than the critical CFL time step, and higher accuracy to handle the non-smoothness of the interface constitutive law (slip weakening friction).The finite element form of the quasi-variational inequality is solved by a Schwarz domain decomposition method, by separating the inner nodes of the domain from the nodes on the fault. In this way, the quasi-variational inequality splits into two subproblems. The first one is a large linear system of equations, and its unknowns are related to the mesh nodes of the first subdomain (i.e. lying inside the domain). The unknowns of the second subproblem are the degrees of freedom of the mesh nodes of the second subdomain (i.e. lying on the domain boundary where the conditions of contact and friction are imposed). This nonlinear subproblem is solved by the same Schwarz algorithm, leading to some local nonlinear subproblems of a very small size.Numerical experiments are performed to illustrate convergence in time and space, instability capturing, energy dissipation and the influence of normal stress variations. We have used the proposed numerical method to compute source dynamics phenomena on complex and realistic 2D fault models (branched fault systems).     

High-order quadratures for the solution of scattering problems in two dimensions

We construct an iterative algorithm for the solution of forward scattering problems in two dimensions. The scheme is based on the combination of high-order quadrature formulae, fast application of integral operators in Lippmann–Schwinger equations, and the stabilized bi-conjugate gradient method (BI-CGSTAB). While the FFT-based fast application of integral operators and the BI-CGSTAB for the solution of linear systems are fairly standard, a large part of this paper is devoted to constructing a class of high-order quadrature formulae applicable to a wide range of singular functions in two and three dimensions; these are used to obtain rapidly convergent discretizations of Lippmann–Schwinger equations. The performance of the algorithm is illustrated with several numerical examples.     

High-speed superplasticity of microcrystalline alloys under conditions of local grain boundary melting

A model is proposed for the high-speed superplasticity of materials under conditions of local grain boundary melting at temperatures close to solidus. It is shown that the local melting of grain boundaries containing segregations of impurity atoms, results in the formation of a structure consisting of liquid-phase regions and solid intergranular bridges which provide cohesion of the grains during the deformation process. The equilibrium concentration, dimensions, and activation energy for the formation of solid bridges are determined as a function of the temperature, initial impurity concentration in the boundary, and the boundary thickness. A mechanism is proposed for grain-boundary slip under conditions of local grain boundary at anomalously high strain rates. Zh. Tekh. Fiz. 68, 38–42 (December 1998)     

A numerical simulation method for dissolution in porous and fractured media

We describe an algorithm for simulating reactive flows in porous media, in which the pore space is mapped explicitly. Chemical reactions at the solid–fluid boundaries lead to dissolution (or precipitation), which makes it necessary to track the movement of the solid–fluid interface during the course of the simulation. We have developed a robust algorithm for constructing a piecewise continuous (C₁) surface, which enables a rapid remapping of the surface to the grid lines. The key components of the physics are the Navier–Stokes equations for fluid flow in the pore space, the convection–diffusion equation to describe the transport of chemical species, and rate equations to model the chemical kinetics at the solid surfaces. A lattice-Boltzmann model was used to simulate fluid flow in the pore space, with linear interpolation at the solid boundaries. A finite-difference scheme for the concentration field was developed, taking derivatives along the direction of the local fluid velocity. When the flow is not aligned with the grid this leads to much more accurate convective fluxes and surface concentrations than a standard Cartesian template. A robust algorithm for the surface reaction rates has been implemented, avoiding instabilities when the surface is close to a grid point. We report numerical tests of different aspects of the algorithm and assess the overall convergence of the method.     

Boundary Conditions for Coupled Quasilinear Wave Equations with Application to Isolated Systems

We consider the initial-boundary value problem for systems of quasilinear wave equations on domains of the form [0, T] × Σ, where Σ is a compact manifold with smooth boundaries ∂Σ. By using an appropriate reduction to a first order symmetric hyperbolic system with maximal dissipative boundary conditions, well posedness of such problems is established for a large class of boundary conditions on ∂Σ. We show that our class of boundary conditions is sufficiently general to allow for a well posed formulation for different wave problems in the presence of constraints and artificial, nonreflecting boundaries, including Maxwell’s equations in the Lorentz gauge and Einstein’s gravitational equations in harmonic coordinates. Our results should also be useful for obtaining stable finite-difference discretizations for such problems.     

An efficient cell-centered diffusion scheme for quadrilateral grids

A new algorithm for solution of diffusion equations in two dimensions on structured quadrilateral grids is proposed. The algorithm is based on a semi-implicit method for the time discretization and has a nine-point local stencil in space. Our scheme is fast, quite accurate and demonstrates good spatial convergence. The presented numerical tests show that it is well suited for hydrocodes with cell-centered principal variables.