Finite-difference outer-layer, analytic inner-layer method for turbulent boundary layers
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Barnwell, Richard W.↗
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Discontinuous Skeletal Gradient Discretisation methods on polytopal meshes
Here, in this work we develop arbitrary-order Discontinuous Skeletal Gradient Discretisations (DSGD) on general polytopal meshes. Discontinuous Skeletal refers to the fact that the globally coupled unknowns are broken polynomials on the mesh skeleton. The key ingredient is a high-order gradient reconstruction composed of two terms: (i) a consistent contribution obtained mimicking an integration by parts formula inside each element and (ii) a stabilising term for which sufficient design conditions are provided. An example of stabilisation that satisfies the design conditions is proposed based on a local lifting of high-order residuals on a Raviart–Thomas–Nédélec subspace. We prove that the novel DSGDs satisfy coercivity, consistency, limit-conformity, and compactness requirements that ensure convergence for a variety of elliptic and parabolic problems. Lastly, links with Hybrid High-Order, non-conforming Mimetic Finite Difference and non-conforming Virtual Element methods are also studied. Numerical examples complete the exposition.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Analysis of iced wings
A method for computing ice shapes along the leading edge of a wing and a method for predicting its aerodynamic performance degradation due to icing is described. Ice shapes are computed using an extension of the LEWICE code which was developed for airfoils. The aerodynamic properties of the iced wing are determined with an interactive scheme in which the solutions of the inviscid flow equations are obtained from a panel method and the solutions of the viscous flow equations are obtained from an inverse three-dimensional finite-difference boundary-layer method. A new interaction law is used to couple the inviscid and viscous flow solutions. The application of the LEWICE wing code to the calculation of ice shapes on a MS-317 swept wing shows good agreement with measurements. The interactive boundary-layer method is applied to a tapered ice wing in order to study the effect of icing on the aerodynamic properties of the wing at several angles of attack.
Cebeci, T.↗
Analysis of iced wings
A method for computing ice shapes along the leading edge of a wing and a method for predicting its aerodynamic performance degradation due to icing is described. Ice shapes are computed using an extension of the LEWICE code which was developed for airfoils. The aerodynamic properties of the iced wing are determined with an interactive scheme in which the solutions of the inviscid flow equations are obtained from a panel method and the solutions of the viscous flow equations are obtained from an inverse three-dimensional finite-difference boundary-layer method. A new interaction law is used to couple the inviscid and viscous flow solutions. The application of the LEWICE wing code to the calculation of ice shapes on a MS-317 swept wing show good agreement with measurements. The interactive boundary layer method is applied to a tapered iced wing in order to study the effect of icing on the aerodynamic properties of the wing at several angles of attack.
Cebeci, Tuncer↗
Recent progress in the analysis of iced airfoils and wings
Recent work on the analysis of iced airfoils and wings is described. Ice shapes for multielement airfoils and wings are computed using an extension of the LEWICE code that was developed for single airfoils. The aerodynamic properties of the iced wing are determined with an interactive scheme in which the solutions of the inviscid flow equations are obtained from a panel method and the solutions of the viscous flow equations are obtained from an inverse three-dimensional finite-difference boundary-layer method. A new interaction law is used to couple the inviscid and viscous flow solutions. The newly developed LEWICE multielement code is amplified to a high-lift configuration to calculate the ice shapes on the slat and on the main airfoil and on a four-element airfoil. The application of the LEWICE wing code to the calculation of ice shapes on a MS-317 swept wing shows good agreement with measurements. The interactive boundary-layer method is applied to a tapered iced wing in order to study the effect of icing on the aerodynamic properties of the wing at several angles of attack.
Cebeci, Tuncer↗
Three-Dimensional High-Order Spectral Finite Volume Method for Unstructured Grids
Many areas require a very high-order accurate numerical solution of conservation laws for complex shapes. This paper deals with the extension to three dimensions of the Spectral Finite Volume (SV) method for unstructured grids, which was developed to solve such problems. We first summarize the limitations of traditional methods such as finite-difference, and finite-volume for both structured and unstructured grids. We then describe the basic formulation of the spectral finite volume method. What distinguishes the SV method from conventional high-order finite-volume methods for unstructured triangular or tetrahedral grids is the data reconstruction. Instead of using a large stencil of neighboring cells to perform a high-order reconstruction, the stencil is constructed by partitioning each grid cell, called a spectral volume (SV), into 'structured' sub-cells, called control volumes (CVs). One can show that if all the SV cells are partitioned into polygonal or polyhedral CV sub-cells in a geometrically similar manner, the reconstructions for all the SVs become universal, irrespective of their shapes, sizes, orientations, or locations. It follows that the reconstruction is reduced to a weighted sum of unknowns involving just a few simple adds and multiplies, and those weights are universal and can be pre-determined once for all. The method is thus very efficient, accurate, and yet geometrically flexible. The most critical part of the SV method is the partitioning of the SV into CVs. In this paper we present the partitioning of a tetrahedral SV into polyhedral CVs with one free parameter for polynomial reconstructions up to degree of precision five. (Note that the order of accuracy of the method is one order higher than the reconstruction degree of precision.) The free parameter will be determined by minimizing the Lebesgue constant of the reconstruction matrix or similar criteria to obtain optimized partitions. The details of an efficient, parallelizable code to solve three-dimensional problems for any order of accuracy are then presented. Important aspects of the data structure are discussed. Comparisons with the Discontinuous Galerkin (DG) method are made. Numerical examples for wave propagation problems are presented.
Liu, Yen↗
Application of mesh refinement to relativistic magnetic reconnection
During relativistic magnetic reconnection, antiparallel magnetic fields undergo a rapid change in topology, releasing a large amount of energy in the form of non-thermal particle acceleration. This work explores the application of mesh refinement to 2D reconnection simulations to efficiently model the inherent disparity in length-scales. We have systematically investigated the effects of mesh refinement and determined necessary modifications to the algorithm required to mitigate non-physical artifacts at the coarse–fine interface. We have used the ultrahigh-order pseudo-spectral analytical time-domain Maxwell solver to analyze how its use can mitigate the numerical dispersion that occurs with the finite-difference time-domain (or “Yee”) method. Absorbing layers are introduced at the coarse–fine interface to eliminate spurious effects that occur with mesh refinement. We also study how damping the electromagnetic fields and current density in the absorbing layer can help prevent the non-physical accumulation of charge and current density at the coarse–fine interface. Using a mesh refinement ratio of 8 for two-dimensional magnetic reconnection simulations, we obtained good agreement with the high-resolution baseline simulation, using only 36% of the macroparticles and 71% of the node-hours needed for the baseline. The methods presented here are especially applicable to 3D systems where higher memory savings are expected than in 2D, enabling comprehensive, computationally efficient 3D reconnection studies in the future.
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
Application of the method of integral relations to unsteady fluid flow problems with shocks
A mixed method using both the method of integral relations and the finite difference technique is developed for the solution of unsteady flow problems. The integral relations method is based upon a chosen interpolating function dependent only on the time domain. The resulting local semidiscrete finite element equations obtained are assembled into a global form. The spatial derivatives at the nodes are replaced by finite difference operators and the discretized nonlinear algebraic system is solved by an iterative scheme. Solutions are obtained for the one-dimensional gasdynamics equation, the one-dimensional wave equation and Burger's model of turbulence. Agreement with other numerical and analytical solutions is excellent for the gasdynamics problem and satisfactory for the Burger equation in cases of small viscous effects.
Wadia, A. R.↗
Domain decomposition methods for systems of conservation laws: Spectral collocation approximations
Hyperbolic systems of conversation laws are considered which are discretized in space by spectral collocation methods and advanced in time by finite difference schemes. At any time-level a domain deposition method based on an iteration by subdomain procedure was introduced yielding at each step a sequence of independent subproblems (one for each subdomain) that can be solved simultaneously. The method is set for a general nonlinear problem in several space variables. The convergence analysis, however, is carried out only for a linear one-dimensional system with continuous solutions. A precise form of the error reduction factor at each iteration is derived. Although the method is applied here to the case of spectral collocation approximation only, the idea is fairly general and can be used in a different context as well. For instance, its application to space discretization by finite differences is straight forward.
Quarteroni, Alfio↗
Gradient-based optimization of 3D MHD equilibria
Using recently developed adjoint methods for computing the shape derivatives of functions that depend on magnetohydrodynamic (MHD) equilibria (Antonsen et al. , J. Plasma Phys. , vol. 85, issue 2, 2019; Paul et al. , J. Plasma Phys. , vol. 86, issue 1, 2020), we present the first example of analytic gradient-based optimization of fixed-boundary stellarator equilibria. We take advantage of gradient information to optimize figures of merit of relevance for stellarator design, including the rotational transform, magnetic well and quasi-symmetry near the axis. With the application of the adjoint method, we reduce the number of equilibrium evaluations by the dimension of the optimization space ( ${\sim }50\text {--}500$ ) in comparison with a finite-difference gradient-based method. We discuss regularization objectives of relevance for fixed-boundary optimization, including a novel method that prevents self-intersection of the plasma boundary. We present several optimized equilibria, including a vacuum field with very low magnetic shear throughout the volume.
Physics↗
Spectral methods for problems in complex geometries
Techniques that permit the efficient application of spectral methods to solve problems in nearly arbitrary geometries are presented. These methods were found to be viable alternatives to finite difference and finite element processes. The spectral methods applied are extensions of the standard techniques of separation of variables to the solution of arbitrarily complicated problems.
Orszag, S. A.↗
Hybrid techniques for complex aerospace electromagnetics problems
Important aerospace electromagnetics problems include the evaluation of antenna performance on aircraft and the prediction and control of the aircraft's electromagnetic signature. Due to the ever increasing complexity and expense of aircraft design, aerospace engineers have become increasingly dependent on computer solutions. Traditionally, computational electromagnetics (CEM) has relied primarily on four disparate techniques: the method of moments (MoM), the finite-difference time-domain (FDTD) technique, the finite element method (FEM), and high frequency asymptotic techniques (HFAT) such as ray tracing. Each of these techniques has distinct advantages and disadvantages, and no single technique is capable of accurately solving all problems of interest on computers that are available now or will be available in the foreseeable future. As a result, new approaches that overcome the deficiencies of traditional techniques are beginning to attract a great deal of interest in the CEM community. Among these new approaches are hybrid methods which combine two or more of these techniques into a coherent model. During the ASEE Summer Faculty Fellowship Program a hybrid FEM/MoM computer code was developed and applied to a geometry containing features found on many modern aircraft.
Aberle, Jim↗
Optimization Based Efficiencies in First Order Reliability Analysis
This paper develops a method for updating the gradient vector of the limit state function in reliability analysis using Broyden's rank one updating technique. In problems that use commercial code as a black box, the gradient calculations are usually done using a finite difference approach, which becomes very expensive for large system models. The proposed method replaces the finite difference gradient calculations in a standard first order reliability method (FORM) with Broyden's Quasi-Newton technique. The resulting algorithm of Broyden updates within a FORM framework (BFORM) is used to run several example problems, and the results compared to standard FORM results. It is found that BFORM typically requires fewer functional evaluations that FORM to converge to the same answer.
Peck, Jeffrey A.↗
FORTRAN program for calculating velocities and streamlines on the hub-shroud mid-channel flow surface of an axial- or mixed-flow turbomachine. 1: User's manual
A FORTRAN 4 computer program has been developed that obtains a subsonic or shock-free transonic flow solution on the hub-shroud mid-channel flow surface of a turbomachine. The blade row may be fixed or rotating, and may be twisted and leaned. Flow may be axial or mixed, up to 45 deg from axial. Upstream and downstream flow variables may vary from hub to shroud, and provision is made to correct for loss of stagnation pressure. The results include velocities, streamlines, and flow angles on the flow surface; and approximate blade surface velocities. Subsonic solutions are obtained by a finite-difference stream-function solution. Transonic solutions are obtained by a velocity-gradient method, using information from a finite-difference stream-function solution at a reduced mass flow.
Katsanis, T.↗
FORTRAN program for calculating velocities and streamlines on the hub-shroud mid-channel flow surface of an axial-or mixed-flow turbomachine. 2: Programmer's manual
A FORTRAN-IV computer program, MERIDL, has been developed that obtains a subsonic or shock-free transonic flow solution on the hub-shroud mid-channel flow surface of a turbomachine. The blade row may be fixed or rotating and may be twisted and leaned. Flow may be axial or mixed, up to 45 deg from axial. Upstream and downstream flow variables can vary from hub to shroud, and provision is made to correct for loss of stagnation pressure. The results include velocities, streamlines, and flow angles on the flow surface and approximate blade surface velocities. Subsonic solutions are obtained by a finite-difference stream-function solution. Transonic solutions are obtained by a velocity-gradient method, using information from a finite-difference stream-function solution at a reduced mass flow.
Katsanis, T.↗
All-optical XNOR gate based on 2D photonic-crystal ring resonators
A novel all-optical XNOR gate is proposed, which combines the nonlinear Kerr effect with photonic-crystal ring resonators (PCRRs). The total size of the proposed optical XNOR gate based on photonic crystals with a square lattice of silicon rods is equal to 35 × 21 μm. The proposed structure has a bandgap in the range from 0.32 to 0.44. To confirm the operation and feasibility of the overall system use is made of analytical and numerical simulation using the dimensional finite difference time domain (FDTD) and plane wave expansion (PWE) methods. (paper)
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
A rapid numerical solution to subsonic flow over planar and axisymmetric profiles at an angle of attack of 0 deg
A numerical solution is presented for the incompressible flow over thin planar and axisymmetric profiles at an angle of attack of 0 degrees. The method uses a finite-difference field solution to the governing equation with a Gauss-Seidel successive overrelaxation scheme. However, the use of a simple Cartesian grid system restricts this method to slender profiles. Results are presented for a cambered airfoil, airfoil in wall effect (two-dimensional flowthrough inlet), body of revolution, and flowthrough nacelle. A computer program is presented which can be used for any of the previously mentioned cases with simple input changes. Results for compressible flow are available with the use of the appropriate two-dimensional or axisymmetric compressibility corrections. Computational time for a typical field calculation of 3000 grid points and 200 cycles through the field is less than 1 minute with less than 50,000 octal storage on the Control Data Corporation 6600 computing system.
Mascitti, V. R.↗
Numerical calculation of transonic boattail flow
A viscid-inviscid interaction procedure for the calculation of subsonic and transonic flow over a boattail was developed. This method couples a finite-difference inviscid analysis with an integral boundary-layer technique. Results indicate that the effect of the boundary layer is as important as an accurate inviscid method for this type of flow. Theoretical results from the solution of the full transonic-potential equation, including boundary layer effects, agree well with the experimental pressure distribution for a boattail. Use of the small disturbance transonic potential equation yielded results that did not agree well with the experimental results even when boundary-layer effects were included in the calculations.
Chow, W. L.↗