Relaxation Regularization Research Articles

Temporally regularized direct numerical simulation

Experience with fluid-flow simulation suggests that, in some instances, under-resolved direct numerical simulation (DNS), without a residual-stress model per se but with artificial damping of small scales to account for energy lost in the cascade from resolved to unresolved scales, may be as reliable as simulations based on more complex models of turbulence. One efficient and versatile manner to selectively damp under-resolved spatial scales is by a relaxation regularization, e.g. Stolz and Adams [S. Stolz, N.A. Adams, An approximate deconvolution procedure for large eddy simulation, Phys. Fluids II (1999) 1699–1701]. We consider the analogous approach based on time scales, time filtering and damping of under-resolved temporal features. The paper explores theoretical and practical aspects of temporally damped fluid-flow simulations. We prove existence of solutions to the resulting continuum model. We also establish the effect of the damping of under-resolved temporal features as the energy balance and dissipation and prove that the time fluctuations → 0 in a precise sense. The method is then demonstrated to obtain both steady-state and time-dependent coarse-grid solutions of the Navier–Stokes equations.

Sparse Tensor Product Wavelet Approximation of Singular Functions

On product domains, sparse-grid approximation yields optimal, dimension-independent convergence rates when the function that is approximated has $L^2$-bounded mixed derivatives of a sufficiently high order. We show that the solution of Poisson's equation on the n-dimensional hypercube with Dirichlet boundary conditions and smooth right-hand side generally does not satisfy this condition. As suggested by P.-A. Nitsche in [Constr. Approx., 21 (2005), pp. 63–81], the regularity conditions can be relaxed to corresponding ones in weighted $L^2$ spaces when the sparse-grid approach is combined with local refinement of the set of one-dimensional wavelet indices towards the end points. In this paper, we prove that for general smooth right-hand sides, the solution of Poisson's problem satisfies these relaxed regularity conditions in any space dimension. Furthermore, since we remove log-factors from the energy-error estimates from Nitsche's work, we show that in any space dimension, locally refined sparse-grid approximation yields the optimal, dimension-independent convergence rate.

Nonmonotonic traveling wave solutions of infiltration into porous media

In uniform soils that are susceptible to unstable preferential flow, the water saturation may exhibit a nonmonotonic profile upon continuous infiltration. As this nonmonotonicity (also known as saturation overshoot) cannot be described by the conventional Richards equation, there have been proposed possible extensions to the unsaturated flow equations, including a nonmonotonic capillary pressure–saturation curve and a second‐order hyperbolic term. Here, we present analytic traveling wave solutions to the extended Richards equation. These new solutions indeed display a nonmonotonic saturation profile, similar to previous simulation results. We show that these extensions need a regularization term to produce a unique solution. We develop complete analytic solutions using a relaxation regularization term, and we discuss the results in terms of recent measurements of saturation overshoot.

Truncation of scales by time relaxation

We study a time relaxation regularization of flow problems proposed and tested extensively by Stolz and Adams. The aim of the relaxation term is to drive the unresolved fluctuations in a computational simulation to zero exponentially fast by an appropriate and often problem dependent choice of its coefficient; this relaxation term is thus intermediate between a tunable numerical stabilization and a continuum modeling term. Our aim herein is to understand how this term, by itself, acts to truncate solution scales and to use this understanding to give insight into parameter selection.

Direction of vorticity and a new regularity criterion for the Navier-Stokes equations

AbstractIn this paper, we prove a new regularity criterion in terms of the direction of vorticity for the weak solution to 3-D incompressible Navier-Stokes equations. Under the framework of Constantin and Fefferman, a more relaxed regularity criterion in terms of the direction of vorticity is established.

Large-Eddy Simulation of Shock-Turbulence Interaction

The approximate deconvolution model (ADM) for large-eddy simulation (LES) is adapted for use with a central fourth-order finite volume (Jameson-type) scheme widely applied for aerodynamics calculations of industrial interest. With ADM, an approximation of the unfiltered solution is obtained by repeated filtering, and given a good approximation of the unfiltered solution the nonlinear terms of the filtered Navier‐Stokes equation are computed directly. The effect of scales not represented on the numerical grid is modeled by a relaxation regularization involving a secondary filter operation and a relaxation parameter. The supersonic turbulent boundary layer along a compression ramp at a freestream Mach number of M =3 and a Reynolds number (based on free-stream quantities and the mean momentum thickness at inflow) of Reθ = 1685 is computed to evaluate the ability of ADM to represent discontinuities and turbulent flow regions. It was observed that a unified modeling of discontinuities and turbulence required a local adaptation of the secondary filter used for the relaxation regularization. The LES results are compared with corresponding filtered direct-numerical-simulation (DNS) data. Very good agreement between the filtered DNS and the LES is observed for the mean, fluctuating, and averaged wall quantities.

The Approximate Deconvolution Model for Large-Eddy Simulation of Compressible Flows With Finite Volume Schemes

The approximate deconvolution model for large-eddy simulation is formulated for a second-order finite volume scheme. With the approximate deconvolution model, an approximation of the unfiltered solution is obtained by repeated filtering, and given a good approximation of the unfiltered solution, the nonlinear terms of the Navier-Stokes equations are computed directly. The effect of scales not represented on the numerical grid is modeled by a relaxation regularization involving a secondary filter operation. A turbulent channel flow at a Mach number of M=1.5 and a Reynolds number based on bulk quantities of Re=3000 is selected for validation of the approximate deconvolution model implementation in a finite volume code. A direct numerical simulation of this configuration has been computed by Coleman et al. Overall, our large-eddy simulation results show good agreement with our filtered direct numerical simulation data. For this rather simple configuration and the low-order spatial discretization, differences between approximate deconvolution model and a no-model computation are found to be small.

The Approximate Deconvolution Model for Large-Eddy Simulation of Compressible Flows With Finite Volume Schemes

The approximate deconvolution model for large-eddy simulation is formulated for a second-order finite volume scheme. With the approximate deconvolution model, an approximation of the unfiltered solution is obtained by repeated filtering, and given a good approximation of the unfiltered solution, the nonlinear terms of the Navier-Stokes equations are computed directly. The effect of scales not represented on the numerical grid is modeled by a relaxation regularization involving a secondary filter operation. A turbulent channel flow at a Mach number of M=1.5 and a Reynolds number based on bulk quantities of Re=3000 is selected for validation of the approximate deconvolution model implementation in a finite volume code. A direct numerical simulation of this configuration has been computed by Coleman et al. Overall, our large-eddy simulation results show good agreement with our filtered direct numerical simulation data. For this rather simple configuration and the low-order spatial discretization, differences between approximate deconvolution model and a no-model computation are found to be small.

The use of LES subgrid‐scale models for shock capturing

AbstractA method for modelling of flow discontinuities based on deconvolution with a relaxation regularization (DDM) is compared with an essentially non‐oscillatory scheme (ENO) without further modelling and with a dynamic Smagorinsky subgrid‐scale model (DSM). For the DDM approach, a sufficiently accurate representation of the filtered non‐linear combination of discontinuous solution components which arise from the convection term is obtained by regularized deconvolution applied to the filtered solution. For stable integration the evolution equations are supplemented by a relaxation regularization based on a secondary filter operation and a relaxation parameter. We apply the above method to the generic test case of a two‐dimensional solenoidal velocity fluctuation field interacting with a normal shock. The results demonstrate a good agreement of DDM with a reference computation, while DDM significantly reduces computational cost. Predictions with DDM are found to provide a more accurate representation of the shock–vorticity–wave interaction phenomena than that which can be accomplished using the ENO scheme with or without subgrid‐scale model. Copyright © 2002 John Wiley & Sons, Ltd.

A Subgrid-Scale Deconvolution Approach for Shock Capturing

We develop a method for the modeling of flow discontinuities which can arise as weak solutions of inviscid conservation laws. Due to its similarity with recently proposed approximate deconvolution models for large-eddy simulation, the method potentially allows for a unified treatment of flow discontinuities and turbulent subgrid scales. A filtering approach is employed since for the filtered evolution equations the solution is smooth and can be solved for by standard central finite-difference schemes without special consideration of discontinuities. A sufficiently accurate representation of the filtered nonlinear combination of discontinuous solution components which arise from the convection term can be obtained by a regularized deconvolution applied to the filtered solution. For stable integration the evolution equations are supplemented by a relaxation regularization based on a secondary filter operation and a relaxation parameter. An estimate for the relaxation parameter is provided. The method is related to the spectral vanishing-viscosity method and the regularized Chapman–Enskog expansion method for conservation laws. We detail the approach and demonstrate its efficiency with the inviscid and viscous Burgers equations, the isothermal shock problem, and the one-dimensional Euler equations.

Electron-emission activity of defects in surface layers of crystalline and vitreous silica

This paper deals with the photoemission activity of radiation defects in crystalline and vitreous silica. Spectral and kinetic parameters and the concentration of defects in the surface layer and the bulk of the material were determined. Some features of the distribution of radiation-induced defects along the depth of samples having different degrees of order were analyzed. Regularities of relaxation of excited states of emission centers suggested that thermally activated processes were involved in the release of electrons.

The approximate deconvolution model for large-eddy simulations of compressible flows and its application to shock-turbulent-boundary-layer interaction

A formulation of the approximate deconvolution model (ADM) for the large-eddy simulation (LES) of compressible flows in complex geometries is detailed. The model is applied to supersonic compression ramp flow where shock-turbulence interaction occurs. With the ADM approach an approximation to the unfiltered solution is obtained from the filtered solution by a series expansion involving repeated filtering. Given a sufficiently good approximation of the unfiltered solution at a time instant, the flux terms of the underlying filtered transport equations can be computed directly, avoiding the need to explicitly compute subgrid-scale closures. The effect of nonrepresented scales is modeled by a relaxation regularization involving a secondary filter operation and a dynamically estimated relaxation parameter. Results of the large-eddy simulation of the turbulent supersonic boundary layer along a compression ramp compare well with filtered DNS data. The filtered shock solution is correctly predicted by the ADM procedure, demonstrating that turbulent and nonturbulent subgrid-scales are properly modeled. We found that a computationally expensive shock-capturing technique was not necessary for stable integration. As a consequence, the computational effort for simulations with ADM is approximately as large as for a coarse-grid DNS with a hybrid compact-upwind-ENO scheme, since the additional computational cost for the subgrid-scale model is more than compensated due to the fact that in the LES flux-derivatives can be computed by linear central finite differences on the entire domain.

An approximate deconvolution model for large-eddy simulation with application to incompressible wall-bounded flows

The approximate deconvolution model (ADM) for the large-eddy simulation of incompressible flows is detailed and applied to turbulent channel flow. With this approach an approximation of the unfiltered solution is obtained by repeated filtering. Given a good approximation of the unfiltered solution, the nonlinear terms of the filtered Navier–Stokes equations can be computed directly. The effect of nonrepresented scales is modeled by a relaxation regularization involving a secondary filter operation. Large-eddy simulations are performed for incompressible channel flow at Reynolds numbers based on the friction velocity and the channel half-width of Reτ=180 and Reτ=590. Both simulations compare well with direct numerical simulation (DNS) data and show a significant improvement over results obtained with classical subgrid scale models such as the standard or the dynamic Smagorinsky model. The computational cost of ADM is lower than that of dynamic models or the velocity estimation model.

Structural relaxation in amorphous substances with highly non-equilibrium structure

Structural relaxation in amorphous substances with highly non-equilibrium structure produced by plasma sputtering, plasma melting, chemical deposition techniques is investigated. It is found that regularities of structural relaxation in these substances are similar to those observed in rapidly quenched glasses and may be described by the Tool–Narayanaswamy glass-transition model. For investigated substances, this result gives an opportunity to predict stability of their properties using a glass-transition model.

Negative norm least-squares methods for the velocity-vorticity-pressure Navier-Stokes equations

We develop and analyze a least-squares finite element method for the steady state, incompressible Navier–Stokes equations, written as a first-order system involving vorticity as new dependent variable. In contrast to standard L2 least-squares methods for this system, our approach utilizes discrete negative norms in the least-squares functional. This allows us to devise efficient preconditioners for the discrete equations, and to establish optimal error estimates under relaxed regularity assumptions. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 237–256, 1999

Jump Conditions in the Non-linear Theory of Thin Irregular Shells

The non-linear theory of thin shell structures with irregularities of geometry, material properties, loading and deformation is developed. The irregular shell is modelled by a reference network being a union of piecewise smooth surfaces and curves, with various fields satisfying relaxed regularity requirements. By transforming the virtual work principle postulated for the entire reference network, we derive the corresponding local field equations and side conditions. Particular attention is paid to formulate the general form of static and kinematic jump conditions at singular geometric and physical curves. Several special kinds of irregularities are considered and some particular forms of the jump conditions are discussed.

Smoothness-relaxation strategies for singular and hypersingular integral equations

Three stages are involved in the formulation of a typical direct boundary element method: derivation of an integral representation; taking a Limit To the Boundary (LTB) so as to obtain an integral equation; and discretization. We examine the second and third stages, focussing on strategies that are intended to permit the relaxation of standard smoothness assumptions. Two such strategies are indicated. Therst is the introduction of various apparent or 'pseudo-LTBs'.The second is 'relaxed regularization', in which a regularized integral equation, derived rigorously under certain smoothness assumptions, is used when less smoothness is available. Both strategies are shown to be based on inconsistent reasoning. Nevertheless, reasons are oered for having some condence in numerical results obtained with certain strategies. Our work is done in two physical contexts, namely two-dimensionalpotential theory (using functions of a complex variable) and three-dimensionalelastostatics. ? 1998 John Wiley & Sons, Ltd.

On the validity of conforming BEM algorithms for hypersingular boundary integral equations

The widely held notion that the use of standard conforming isoparametric boundary elements may not be used in the solution of hypersingular integral equations is investigated. It is demonstrated that for points on the boundary where the underlying field is C1,α continuous, a class of rigorous nonsingular conforming BEM algorithms may be applied. The justification for this class of algorithms is interpreted in terms of some recent criticism. It is shown that the numerical integration in these conforming BEM algorithms using relaxed regularization represents a finite approximation to the standard two-sided Hadamard finite part interpretation of hypersingular integrals. It is also shown that the integration schemes in this class of algorithms are not based upon one-sided finite part interpretations. As a result, the attendant ambiguities associated with the incorrect use of the one-sided interpretations in boundary integral equations pose no problem for this class of algorithms. Additionally, the distinction is made between the analytic discontinuities in the field which place limitations on the applicability of the conforming BEM and the discontinuities resulting from the use of piece-wise C1,α interpolations.

Finite element analysis of smooth, folded and multi-shell structures

The paper is concerned with the nonlinear theory and finite element analysis of shell structures with an arbitrary geometry, loading and boundary conditions. A complete set of shell field equations and side conditions (boundary and jump conditions) is derived from the basic laws of continuum mechanics. The developed shell theory includes the so-called drilling couples as well as the drilling rotation. It is shown that this property is crucial in the analysis of irregular shell structures, such as those containing folds, branches, column supports, stiffeners, etc. The relevant variational principles with relaxed regularity requirements are also presented. These principles provide the mathematical basis for the formulation of various classes of shell finite elements. The developed finite elements include a displacement/rotation based Lagrange family, a stress resultant based mixed and a semi-mixed family as well as so-called assumed strain elements. All elements have six degrees of freedom at each node, three translational and three rotational ones, including the drilling rotation formulated on the foundation of an exact (in defined sense) shell theory. As such, they are equally applicable to smooth as well as to irregular shell structures. The general applicability of the developed elements is illustrated through an extensive numerical analysis of the representative test examples. In order to obtain a still deeper insight into the problem a Lagrange family of standard degenerated shell elements with five degrees of freedom per node and an element with six degrees of freedom per node based on the von Karman plate theory are considered as well. The presented numerical results include complex plate and doubly-curved shell structures. Linear and non-linear solutions with a pre- and post-buckling analysis are discussed.

Relaxation kinetics of correlated neutral Frenkel pairs of defects in alkali-halide crystals

A modelling method is developed that permits qualitative investigation of the evolution of genetic pairs of Frenkel defects in alkali-halide crystals (AHC). The relaxation kinetics and the form of the low-temperature annealing curves of F-, H-pairs are determined by initial mutual distribution functions (IDF) of the pairs being formed up to the time of excess energy dissipation by an H-center. It is shown that, because of discreteness of the lattice for AHC, interactions between the F- and H-centers are characteristic of IDF with a nonuniform spacing distribution of the F- and H-pairs. Fundamental regularities of Frenkel defect relaxation after pulse exposure and of F-, H-pair annealing after low-temperature AHC exposure are discussed on the basis of modelling results.