Implicit Operator Research Articles

Upwind differencing and LU factorization for chemical non-equilibrium navier-stokes equations

An efficient and robust upwind method for solving the chemical non-equilibrium Navier-Stokes equations has been developed. The method uses either the Roe or Van Leer flux-splitting for inviscid terms and central differencing for viscous terms in the explicit operator (residual), and the Steger-Warming (SW) splitting and lower-upper (LU) approximate factorization for the implicit operator. This approach is efficient since the SW-LU combination requires the inversion of only block diagonal matrices, as opposed to the block tridiagonal inversion of the widely used ADI method, and is fully vectorizable. The LU method is particularly advantageous for systems with large number of equations, such as for chemical and thermal nonequilibrium flow. Formulas of the numerical method are presented for the finite-volume discretization of the Navier-Stokes equations in general coordinates. Numerical tests in hypersonic blunt body, ramped-duct, shock wave/boundary layer interaction, and divergent nozzle flows demonstrate the efficiency and robustness of the present method.

A Fully Implicit Quine-McCluskey Procedure Using BDD’s

We present an exact method for minimizing logic functions using BDD's to represent our functions. This approach differs from the classical approach in that it exploits the properties of the BDD data structure and the properties of a new extended space that we define, in order to implicitly compute the Primes, Minterms and Covering table for the Quine-McCluskey procedure. In this method the function is mapped to an extended space which endows it with special properties that may be exploited to compute the function Primes and Minterms. The next step consists of conceptually creating a covering table whose rows represent the minterms and whose columns represent the primes. We formulate conditions for row and column dominance and remove dominated rows and columns iteratively until no more reduction is possible. The final step consists of finding a minimum column cover for the remaining cyclic core of the problem. All functions are implemented using implicit BDD operations.

A practical implementation of turbulence models for the computation of three‐dimensional separated flows

AbstractAn upwind MUSCL‐type implicit scheme for the three‐dimensional Navier‐Stokes equations is presented and details on the implementation for three‐dimensional flows of a ‘diagonal’ upwind implicit operator are developed. Turbulence models for separated flows are also described with an emphasis on the numerical specificities of the Johnson‐King non‐equilibrium model. Good predictions of separated two‐ and three‐dimensional flows are demonstrated.

ON DIRECT PRODUCT DECOMPOSITIONS OF FINITE $\mathscr J$-TRIVIAL SEMIGROUPS

A nontrivial decomposition of the class J of all finite [Formula: see text]-trivial semigroups as a join of two pseudovarieties is given. The definition of these pseudovarieties and the proof are purely syntactical, being based on a complete description of the implicit operations on J obtained by the author in a previous paper.

On finite simple semigroups

Implicit operations (new operations commuting with all old homomorphisms) on pseudovarieties have been shown to play an important role in the study of these classes. They may be used to axiomatize sub-pseudovariaties and to describe recognizable subsets of (relatively) free objects. This paper presents a case study for the pseudovariety CS consisting of all finite simple semigroups. Based on a result of profinite group theory, a structural description of semigroups of implicit operations on finite simple semigroups is used to deduce that CS is join-irreducible.

Acceleration of convergence by shifting the spectrum of implicit finite difference operators associated with the equations of gas dynamics

AbstractEigensystem analysis techniques are applied to finite difference formulations of the Navier‐Stokes equations in one dimension. Spectra of the resulting implicit difference operators are computed. The largest eigenvalues are calculated by using a combination of the Frechet derivative of the operators and Arnoldi's method. The accuracy of Arnoldi's method is tested by comparing the rate of convergence of the iterative method with the dominant eigenvalue of the original iteration matrix.On the basis of the pattern of eigenvalue distributions for various flow configurations, a shifting of the implicit operators in question is devised. The idea of shifting is based on the power method of linear algebra and is very simple to implement. This procedure has improved the rates of convergence of CFD codes (developed at NASA Ames Research Center) by 20%–50%. The sensitivity of the computed solution with respect to the shift is also studied. Finally, an adaptive shifting of the spectrum together with Wynn's acceleration algorithm are discussed. It turns out that the shifting process is a preconditioner for Wynn's method.

A numerical method for the vorticity-velocity Navier-Stokes equations in two and three dimensions

Abstract This paper provides a numerical method for solving the steady-state vorticity-velocity Navier-Stokes equations in two and three dimensions. The vorticity transport equation is considered together with a Poisson equation for the velocity vector, the latter equation being parabolized in time according to the false transient approach. The two vector equations are discretized in time using the implicit Euler time stepping and the delta form of Beam and Warming. A staggered-grid spatial discretization is employed in conjunction with a deferred correction procedure. Second-order-accurate central differences are used to approximate the steady-state residuals, written in conservative form for accuracy reasons, whereas upwind differences are used for the advection terms in the implicit operator, to obtain diagonally-dominant tridiagonal systems. The discrete equations are solved sequentially by means of a robust alternating direction line-Gauss-Seidel iteration procedure combined with a simple multigrid strategy. For the model driven-cavity-flow problem in two and three dimensions, the method is found to be efficient and very accurate. For the first time, the three-dimensional discrete vorticity and velocity fields, computed using a Poisson equation for the velocity vector, are both solenoidal and satisfy their mutual relationship, exactly.

Implicit operations on finite [formula omitted]-trivial semigroups and a conjecture of I. Simon

Let J be the pseudovariety of all finite J -trivial semigroups and let Ω̄ n J denote the topological semigroup of all n-ary implicit operations on J. The semigroup Ω̄ n J is generated by the n component projections together with the 2 n −1 idempotents. Furthermore, Ω̄ n J is described as a free algebra of type (1,2) in a certain variety and the word problem is solved in this algebra. As a consequence, Ω̄ n J is countable, which settles a conjecture proposed by I. Simon.

Multigrid solution of the Navier-Stokes equations on triangular meshes

A Navier-Stokes algorithm for use on unstructured triangular meshes is presented. Spatial discretization of the governing equations is achieved using a finite element Galerkin approximation, which can be shown to be equivalent to a finite volume approximation for regular equilateral triangular meshes. Integration steady-state is performed using a multistage time-stepping scheme, and convergence is accelerated by means of implicit residual smoothing and an unstructured multigrid algorithm. Directional scaling of the artificial dissipation and the implicit residual smoothing operator is achieved for unstructured meshes by considering local mesh stretching vectors at each point. The accuracy of the scheme for highly stretched triangular meshes is validated by comparing computed flat-plate laminar boundary layer results with the well known similarity solution, and by comparing laminar airfoil results with those obtained from various well-established structured quadrilateral-mesh codes. The convergence efficiency of the present method is also shown to be competitive with those demonstrated by structured quadrilateral-mesh algorithms.

The category of complete lattices as a category of algebras

Complete lattices are regarded as algebras whose operations are joins and meets of arbitary arities. We show that, unlike algebras of classical algebraic theories, they admit implicit operations (operations compatible with homomorphisms) which are not induced by terms and that there are more than a proper class of such “wild” operations. We show how they are related to terms.

String potential model: Spinless quarks.

We propose a generalization of the usual potential model of quark bound states in which the confining flux tube is a dynamical object carrying both angular momentum and energy. The QQ\ifmmode\bar\else\textasciimacron\fi{}-string system with spinless quarks is quantized using an implicit operator technique and the resulting relativistic wave equation is solved. For heavy quarks the usual Schr\"odinger valence-quark model is recovered. The Regge slope with light quarks agrees with the classical rotating-string result and is significantly larger than that given by the normal potential model. Quark mass variation and the effects of short-range forces are also considered.

The algebra of implicit operations

For an ordinalα and a class\(C\) of topological algebras of a given type (which may be infinite and may contain inflnitary operations), anα-aryimplicit operation on\(C\) is any “new”α-ary operation whose introduction does not eliminate any continuous homomorphisms between members of\(C\). The set of allα-ary implicit operations on\(C\) is denoted by\(\bar \Omega _\alpha C\) and forms an algebra of the given type which is endowed with the least topology making continuous all homomorphisms into members of\(C\). With this topology,\(\bar \Omega _\alpha C\) is a topological algebra in which the subalgebraΩ α \(C\) of allα-ary operations on\(C\) which are induced by terms is dense, provided that\(C\) is closed under the formation of closed subalgebras and finitary direct products. This is obtained by realizing\(\bar \Omega _\alpha C\) as an inverse limit ofα-generated members of\(C\). These results are applied to pseudovarieties of topological and finite algebras.

More thoughts on the unconscious: Reply to Brody and to Lewicki and Hill.

Brody (1989) and Lewicki and Hill (1989) present commentaries on my central thesis (Reber, 1989) that there exist powerful induction routines that operate largely independently of awareness and yield rich and complex tacit knowledge that resists attempts to make it conscious, although these commentaries are of dramatically different kinds. Because there is a fairly clear basis of agreement between Lewicki and Hill and the points that I made, this reply deals almost entirely with the issues raised by Brody. The focus is on two classes of issues: (a) those of a methodological nature that surround the general problem of doing experiments on unconscious cognition and (b) those that derive from considerations of evolutionary biology that provide a basis for arguing that implicit operations are primary and form the foundation for conscious processes.

A new approximate LU factorization scheme for the Reynolds-averaged Navier-Stokes equations

A new approximate LU factorization scheme is developed to solve the steady state Reynolds-averaged Navier-Stokes (NS) equations. Central differencing is used for both implicit and explicit operator and special care is taken to obtain well-conditioned factors on the implicit side. The scheme is then analyzed and optimized according to a simple linear analysis. It is unconditionally stable for the model hyperbolic equation in both two- and three-dimensions. However, the requirement for well-conditioned factors has essentially limited the effective time step the scheme can achieve. Supersonic and transonic three-dimensinal flows past a hemisphere cylinder are computed to demonstrate the convergence characteristics of the scheme. A good convergence rate is achieved for the inviscid case. Finally, an explicit eigenvector annihilation procedure is adopted successfully to remove the stiffness caused by the fine grid spacing for viscous flows.

Effects of various implicit operators on a flux vector splitting method

Three different implicit operators in a numerical method for solving the two-dimensional steady Euler equations using flux vector splitting are investigated. These include the implementation of the true Jacobian matrices in the implicit part, the use of their splitting approximate form, and a modification of the implicit part of the scheme that uses the scalar form of the implicit part based on the spectral radii of the split Jacobian matrices. The three versions of the basic algorithm are compared with the results given by two common numerical methods using several two-dimensional test cases. Special attention is paid to the quality of results as well as computational efficiency and convergence properties.

Implicit TVD schemes for hyperbolic conservation laws in curvilinearcoordinates

The Harten (1983, 1984) total variation-diminishing (TVD) schemes, constituting a one-parameter explicit and implicit, second-order-accurate family, have the property of not generating spurious oscillations when applied to one-dimensional, nonlinear scalar hyperbolic conservation laws and constant coefficient hyperbolic systems. These methods are presently extended to the multidimensional hyperbolic conservation laws in curvilinear coordinates. Means by which to linearize the implicit operator and solution strategies, in order to improve the computation efficiency of the implicit algorithm, are discussed. Numerical experiments with steady state airfoil calculations indicate that the proposed linearized implicit TVD schemes are accurate and robust.

Linearized form of implicit TVD schemes for the multidimensional Euler and Navier-Stokes equations

Linearized alternating direction implicit (ADI) forms of a class of total vvariation diminishing (TVD) schemes for the Euler and Navier-Stokes equations have been developed. These schemes are based on the second-order-accurate TVD schemes for hyperbolic conservation laws developed by Harten[1,2]. They have the property of not generating spurious oscillations across shocks and contact discontinuities. In general, shocks can be captured within 1–2 grid points. These schemes are relatively simple to understand and easy to implement into a new or existing computer code. One can modify a standard three-point central-differences code by simply changing the conventional numerical dissipation term into the one designed for the TVD scheme. For steady-state applications, the only difference in computation is that the current schemes require a more elaborate dissipation term for the explicit operator; no extra computation is required for the implicit operator. Numerical experiments with the proposed algorithms on a variety of steady-state airfoil problems illustrate the versatility of the schemes.

Three-dimensional calculation of transonic viscous flows by an implicit method

A new implicit finitevolume method for solving the three-dimensional Navier-Stokes equations for compressible fluids is presented and applied to the calculation of flows over swept wings at moderate Reynolds numbers. The method makes use of an explicit predictor-corrector scheme to calculate a provisional value, which is then corrected by using an implicit operator. The explicit part is a generalization of the centered Thornmen scheme, while the implicit operator is a combination of the inviscid schemes proposed by Lerate and the viscous treatment of Beam and Warming. Results are here presented for a swept wing between two parallel walls and for the finite wing ONERA M6 at Mx =0.8, zero incidence and Re^ = 1000. The computations are performed with a time-step about 5 to 10 times greater than the explicit step.

A factored implicit scheme for numerical weather prediction

AbstractThe nonlinear partial differential equations of atmospheric dynamics govern motion on two time scales, a fast one and a slow one. Only the slow‐scale motions are relevant in predicting the evolution of large weather patterns. Implicit numerical methods are therefore attractive for weather prediction, since they permit a large time step chosen to resolve only the slow motions.To develop an implicit method which is efficient for problems in more than one spatial dimension, one must approximate the problem by smaller, usually one‐dimensional problems. A popular way to do so is to approximately factor the multidimensional implicit operator into one‐dimensional operators. The factorization error incurred in such methods, however, is often unacceptably large for problems with multiple time scales.We propose a new factorization method for numerical weather prediction which is based on factoring separately the fast and slow parts of the implicit operator. We show analytically that the new method has small factorization error, which is comparable to other discretization errors of the overall scheme. The analysis is based on properties of the shallow water equations, a simple two‐dimensional version of the fully three‐dimensional equations of atmospheric dynamics.

Solution algorithms for nonlinear transient heat conduction analysis employing element-by-element iterative strategies

Despite continuing advancements in computer technology, there are many problems of engineering interest that exceed the combined capabilities of today's numerical algorithms and computational hardware. The resources required by traditional finite element algorithms tend to grow geometrically as the problem size is increased. Thus, for the forseeable future, there will be problems of interest which cannot be adequately modeled using currently available algorithms. For this reason, we have undertaken the development of algorithms whose resource needs grow only linearly with problem size. In addition, these new algorithms will fully exploit the ‘parallel-processing’ capability available in the new generation of multi-processor computers. The approach taken in the element-by-element solution procedures is to approximate the global implicit operator by a product of lower-order operators. This type of product approximation originated with ADI techniques and was further refined into the ‘method of fraction steps’. The current effort involves the use of a more natural operator split for finite element analysis based on element operators. This choice of operator splitting based on element operators has several advantages. First, it fits easily within the architecture of current finite element programs. Second, it allows the development of ‘parallel’ algorithms. Finally, the computational expense varies only linearly with the number of elements. The particular problems considered arise from nonlinear transient heat conduction. The nonlinearity enters through both material temperature dependence and radiation boundary conditions. The latter condition typically introduces a ‘stiff’ component in the resultant matrix ODEs which precludes effective use of explicit solution techniques. On the other hand, typical implicit techniques involving direct equation solving can be prohibitively expensive. Instead, the matrix equations are solved by combining a modified Newton-Raphson iteration scheme for the nonlinear algebraic problem with an element-by-element preconditioned conjugate gradient procedure for the linear problems. An adaptive intercommunicative strategy is developed for setting termination criteria for the nonlinear and linear equation solving algorithms. Errors and computational efficiency in the transient solution are controlled by an automatic time-step selection strategy. The resultant procedure has proven to be both accurate and reliable in the solution of medium-size test problems.