Donald W. Schwendeman

The Riemann problem and a high-resolution Godunov method for a model of compressible two-phase flow

This paper considers the Riemann problem and an associated Godunov method for a model of compressible two-phase flow. The model is a reduced form of the well-known Baer-Nunziato model that describes the behavior of granular explosives. In the analysis presented here, we omit source terms representing the exchange of mass, momentum and energy between the phases due to compaction, drag, heat transfer and chemical reaction, but retain the non-conservative nozzling terms that appear naturally in the model. For the Riemann problem the effect of the nozzling terms is confined to the contact discontinuity of the solid phase. Treating the solid contact as a layer of vanishingly small thickness within which the solution is smooth yields jump conditions that connect the states across the contact, as well as a prescription that allows the contribution of the nozzling terms to be computed unambiguously. An iterative method of solution is described for the Riemann problem, that determines the wave structure and the intermediate states of the flow, for given left and right states. A Godunov method based on the solution of the Riemann problem is constructed. It includes non-conservative flux contributions derived from an integral of the nozzling terms over a grid cell. The Godunov method is extended to second-order accuracy using a method of slope limiting, and an adaptive Riemann solver is described and used for computational efficiency. Numerical results are presented, demonstrating the accuracy of the numerical method and in particular, the accurate numerical description of the flow in the vicinity of a solid contact where phases couple and nozzling terms are important. The numerical method is compared with other methods available in the literature and found to give more accurate results for the problems considered.

Two‐Dimensional Wafer‐Scale Chemical Mechanical Planarization Models Based on Lubrication Theory and Mass Transport

The effects of the variation of chemical mechanical planarization (CMP) process parameters on slurry hydrodynamics and removal rate are studied using physically based models. The two models which are developed to describe and fundamentally understand the CMP process are (i) the lubrication model for slurry flow and ( ii) the mass transport model for material removal. The mass transport model is developed for copper CMP. Conditions for stable operation and reduced wafer scratching are identified from the lu brication model. The mass transport model takes into account the chemical reaction at the wafer surface, the slurry flow hydrodynam ics, and the presence of abrasive particles. The polish rates predicted by the model agree well with those measured experimental ly.

Moving overlapping grids with adaptive mesh refinement for high-speed reactive and non-reactive flow

We consider the solution of the reactive and non-reactive Euler equations on two-dimensional domains that evolve in time. The domains are discretized using moving overlapping grids. In a typical grid construction, boundary-fitted grids are used to represent moving boundaries, and these grids overlap with stationary background Cartesian grids. Block-structured adaptive mesh refinement (AMR) is used to resolve fine-scale features in the flow such as shocks and detonations. Refinement grids are added to base-level grids according to an estimate of the error, and these refinement grids move with their corresponding base-level grids. The numerical approximation of the governing equations takes place in the parameter space of each component grid which is defined by a mapping from (fixed) parameter space to (moving) physical space. The mapped equations are solved numerically using a second-order extension of Godunovs method. The stiff source term in the reactive case is handled using a Runge-Kutta error-control scheme. We consider cases when the boundaries move according to a prescribed function of time and when the boundaries of embedded bodies move according to the surface stress exerted by the fluid. In the latter case, the Newton-Euler equations describe the motion of the center of mass of the each body and the rotation about it, and these equations are integrated numerically using a second-order predictor-corrector scheme. Numerical boundary conditions at slip walls are described, and numerical results are presented for both reactive and non-reactive flows that demonstrate the use and accuracy of the numerical approach.

Pad porosity, compressibility and slurry delivery effects in chemical- mechanical planarization: modeling and experiments

A chemical-mechanical planarization (CMP) model based on lubrication theory is developed which accounts for pad compressibility, pad porosity and means of slurry delivery. Slurry film thickness and velocity distributions between the pad and the wafer are predicted using the model. Two regimes of CMP operation are described: the lubrication regime (for ,40‐70 mm slurry film thickness) and the contact regime (for thinner films). These regimes are identified for two different pads using experimental copper CMP data and the predictions of the model. The removal rate correlation based on lubrication and mass transport theory agrees well with our experimental data in the lubrication regime. q 2000 Elsevier Science S.A. All rights reserved.

An adaptive numerical scheme for high-speed reactive flow on overlapping grids

We describe a method for the numerical solution of high-speed reactive flow in complex geometries using overlapping grids and block-structured adaptive mesh refinement. We consider flows described by the reactive Euler equations with an ideal equation of state and various stiff reaction models. These equations are solved using a second-order accurate Godunov method for the convective fluxes and a Runge-Kutta time-stepping scheme for the source term modeling the chemical reactions. We describe an extension of the adaptive mesh refinement approach to curvilinear overlapping grids. Numerical results are presented showing the evolution to detonation in a quarter-plane provoked by a temperature gradient and the propagation of an overdriven detonation in an expanding channel. The first problem, which considers a one-step Arrhenius reaction model, is used primarily to validate the numerical method, while the second problem, which considers a three-step chain-branching reaction model, is used to illustrate mechanisms of detonation failure and rebirth for the channel geometry.

Mechanisms of detonation formation due to a temperature gradient

Emergence of a detonation in a homogeneous, exothermically reacting medium can be deemed to occur in two phases. The first phase processes the medium so as to create conditions ripe for the onset of detonation. The actual events leading up to preconditioning may vary from one experiment to the next, but typically, at the end of this stage the medium is hot and in a state of nonuniformity. The second phase consists of the actual formation of the detonation wave via chemico-gasdynamic interactions. This paper considers an idealized medium with simple, rate-sensitive kinetics for which the preconditioned state is modelled as one with an initially prescribed linear gradient of temperature. Accurate and well-resolved numerical computations are carrried out to determine the mode of detonation formation as a function of the size of the initial gradient. For shallow gradients, the result is a decelerating supersonic reaction wave, a weak detonation, whose trajectory is dictated by the initial temperature profile, with only weak intervention from hydrodynamics. If the domain is long enough, or the gradient less shallow, the wave slows down to the Chapman–Jouguet speed and undergoes a swift transition to the ZND structure. For sharp gradients, gasdynamic nonlinearity plays a much stronger role. Now the path to detonation is through an accelerating pulse that runs ahead of the reaction wave and rearranges the induction-time distribution there to one that bears little resemblance to that corresponding to the initial temperature gradient. The pulse amplifies and steepens, transforming itself into a complex consisting of a lead shock, an induction zone, and a following fast deflagration. As the pulse advances, its three constituent entities attain progressively higher levels of mutual coherence, to emerge as a ZND detonation. For initial gradients that are intermediate in size, aspects of both the extreme scenarios appear in the path to detonation. The novel aspect of this study resides in the fact that it is guided by, and its results are compared with, existing asymptotic analyses of detonation evolution.

Parallel computation of three-dimensional flows using overlapping grids with adaptive mesh refinement

This paper describes an approach for the numerical solution of time-dependent partial differential equations in complex three-dimensional domains. The domains are represented by overlapping structured grids, and block-structured adaptive mesh refinement (AMR) is employed to locally increase the grid resolution. In addition, the numerical method is implemented on parallel distributed-memory computers using a domain-decomposition approach. The implementation is flexible so that each base grid within the overlapping grid structure and its associated refinement grids can be independently partitioned over a chosen set of processors. A modified bin-packing algorithm is used to specify the partition for each grid so that the computational work is evenly distributed amongst the processors. All components of the AMR algorithm such as error estimation, regridding, and interpolation are performed in parallel. The parallel time-stepping algorithm is illustrated for initial-boundary-value problems involving a linear advection-diffusion equation and the (nonlinear) reactive Euler equations. Numerical results are presented for both equations to demonstrate the accuracy and correctness of the parallel approach. Exact solutions of the advection-diffusion equation are constructed, and these are used to check the corresponding numerical solutions for a variety of tests involving different overlapping grids, different numbers of refinement levels and refinement ratios, and different numbers of processors. The problem of planar shock diffraction by a sphere is considered as an illustration of the numerical approach for the Euler equations, and a problem involving the initiation of a detonation from a hot spot in a T-shaped pipe is considered to demonstrate the numerical approach for the reactive case. For both problems, the accuracy of the numerical solutions is assessed quantitatively through an estimation of the errors from a grid convergence study. The parallel performance of the approach is examined for the shock diffraction problem.

A high-resolution Godunov method for compressible multi-material flow on overlapping grids

A numerical method is described for inviscid, compressible, multi-material flow in two space dimensions. The flow is governed by the multi-material Euler equations with a general mixture equation of state. Composite overlapping grids are used to handle complex flow geometry and block-structured adaptive mesh refinement (AMR) is used to locally increase grid resolution near shocks and material interfaces. The discretization of the governing equations is based on a high-resolution Godunov method, but includes an energy correction designed to suppress numerical errors that develop near a material interface for standard, conservative shock-capturing schemes. The energy correction is constructed based on a uniform-pressure-velocity flow and is significant only near the captured interface. A variety of two-material flows are presented to verify the accuracy of the numerical approach and to illustrate its use. These flows assume an equation of state for the mixture based on the Jones-Wilkins-Lee (JWL) forms for the components. This equation of state includes a mixture of ideal gases as a special case. Flow problems considered include unsteady one-dimensional shock-interface collision, steady interaction of a planar interface and an oblique shock, planar shock interaction with a collection of gas-filled cylindrical inhomogeneities, and the impulsive motion of the two-component mixture in a rigid cylindrical vessel.

A theory of pad conditioning for chemical-mechanical polishing

Statistical models are presented to describe the evolution of the surface roughness of polishing pads during the pad-conditioning process in chemical-mechanical polishing. The models describe the evolution of the surface-height probability-density function of solid pads during fixed height or fixed cut-rate conditioning. An integral equation is derived for the effect of conditioning on a foamed pad in terms of a model for a solid pad. The models that combine wear and conditioning are then discussed for both solid and foamed pads. Models include the dependence of the surface roughness on the shape and density of the cutting tips used in the conditioner and on other operating parameters. Good agreement is found between the model, Monte Carlo simulations and with experimental data.

On converging shock waves

An approximate theory of shock dynamics is used to study the behaviour of converging cylindrical shocks. For cylindrical shocks with regular polygonal-shaped cross sections, exact solutions are found, showing that an original polygonal shape repeats at successive intervals with successive contractions in scale. In this sense, these shapes are stable, and the successive Mach numbers increase according to exactly the same formula as for a circular cylindrical shock. The behaviour for initial shock shapes close to these and the general tendency of perturbed circular shapes to become polygonal, not necessarily regular, is explored numerically. Further analytical results are provided for rectangular shapes. Comments are made on the interpretation of regular reflection in this theory and on converging shocks in three dimensions.