Parametric spline “mesh” generation for nuclear engineering
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22 GENERAL STUDIES OF NUCLEAR REACTORS↗
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Turbomachine with internal bearing and rotor-spline interface cooling and systems incorporating the same
Aspects of the present disclosure include turbomachines designed and configured for high temperature and pressure operation and increased power level output that minimize pressure vessel design requirements, and increase dry gas seal reliability. In some examples, a first radial bearing is located in a high temperature and/or pressure region of the turbomachine between a rotor of the machine and a dry gas seal while other bearings are located outside of the high pressure region.
Fairman, Kevin D.↗
Spline-Based Electromagnetic Simulation on HPC Platforms
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97 MATHEMATICS AND COMPUTING↗
Quick acting gimbal joint
An improved linkage assembly provides for locking the pivotable position of one member relative to another member. The linkage assembly permits the unlocking of these members, their subsequent repositioning, and the reliable relocking in a desired position without utilizing tooling. The first member includes a pair of spaced apart support flanges each having a through aperture therein defining a flange female spline. The second member has an end positioned between the pair of support flanges and also includes a through aperture defining a pair of second member female splines. A pair of control members are each slidably moveable along the common axis and extend through the respective flanges. Each control member has a male splined portion which normally engages both the respective flange female spline and the second member female spline to lock the rotatable position of the first and second members. When each of the control members are pressed inward simultaneously, the male splined portions disengage the flange female splines to permit rotation of the first member about the common axis with respect to the second member.
William B Wood↗
An arbitrarily high-order three-dimensional Cartesian-grid method for reconstructing interfaces from volume fraction fields
Here Tthis work describes a newly developed, arbitrarily high-order Cartesian-grid method for reconstructing material interfaces from a volume fraction field. The method begins by identifying all of the grid cells in the volume fraction field that are intersected by the interface and need to be approximated by the reconstruction scheme. Finite-differences are used to calculate the gradient of the volume fraction field and provide an estimate of the surface normal in all of the interfacial grid cells. Groups of connected grid cells are then identified which all have the same dominant component of the normal vector. This grouping by orientation determines the proper dependent variable to use in the surface reconstruction (e.g. for a 2D curve, this step determines if the surface will be approximated by a function of x or y). A cumulative integral over the surface is constructed and fit using b-splines for two-dimensional problems or tensor-product b-splines for three-dimensional problems. This construction allows for the interface to be recovered through application of the second fundamental theorem of calculus. Fitting the cumulative integral with $\mathscr{N}$ th-order b-splines (or tensor-product b-splines) yields an ($\mathscr{N}$-1) th-order convergence rate of the interface shape. Differentiation of the b-spline interface function(s) allows for the high-order approximation of the normal vector and curvature to be obtained directly anywhere along b-spline. Together, the proposed reconstruction technique can achieve arbitrarily high mesh convergence rates. Validation tests are presented with mesh convergence rates ranging from fourth- to tenth-order.
97 MATHEMATICS AND COMPUTING↗
Hierarchical Control and Trajectory Planning
Most of the time on this project was spent on the trajectory planning problem. The construction is equivalent to the classical spline construction in the case that the system matrix is nilpotent. If the dimension of the system is n then the spline of degree 2n-1 is constructed. This gives a new approach to the construction of splines that is more efficient than the usual construction and at the same time allows the construction of a much larger class of splines. All known classes of splines are reconstructed using the approach of linear control theory. As a numerical analysis tool control theory gives a very good tool for constructing splines. However, for the purposes of trajectory planning it is quite another story. Enclosed in this document are four reports done under this grant.
Martin, Clyde F.↗
Adaptive immersed isogeometric level-set topology optimization
Here, this paper presents for the first time an adaptive immersed approach for level-set topology optimization using higher-order truncated hierarchical B-spline discretizations for design and state variable fields. Boundaries and interfaces are represented implicitly by the iso-contour of one or multiple level-set functions. An immersed finite element method, the eXtended IsoGeometric Analysis, is used to predict the physical response. The proposed optimization framework affords different adaptively refined higher-order B-spline discretizations for individual design and state variable fields. The increased continuity of higher-order B-spline discretizations together with local refinement enables direct control over the accuracy of the representation of each field while simultaneously reducing computational cost compared to uniformly refined discretizations. A flexible mesh adaptation strategy enables local refinement based on geometric measures or physics-based error indicators. These adaptive discretization and analysis approaches are integrated into gradient-based optimization schemes, evaluating the design sensitivities using the adjoint method. Numerical studies illustrate the features of the proposed framework with static, linear elastic, multi-material, two- and three-dimensional problems. The examples provide insight into the effect of refining the design variable field on the optimization result and the convergence rate of the optimization process. Using coarse higher-order B-spline discretizations for level-set fields promotes the development of smooth designs and suppresses the emergence of small features. Moreover, adaptive mesh refinement for state variable fields results in a reduction of overall computational cost. Higher-order B-spline discretizations are especially interesting when evaluating gradients of state variable fields due to their higher inter-element continuity.
36 MATERIALS SCIENCE↗
Polynominal Interpolation Methods for Viscous Flow Calculations
Higher-order collocation procedures resulting in tridiagonal matrix systems are derived from polynomial spline interpolation and by Hermitian (Taylor series) finite-difference discretization. The similarities and special features of these different developments are discussed. The governing systems apply for both uniform and variable meshes. Hybrid schemes resulting from two different polynomial approximations for the first and second derivatives lead to a nonuniform mesh extension of the so-called compact or Pad? difference technique (Hermite 4). A variety of fourth-order methods are described and the Hermitian approach is extended to sixth-order (Hermite 6). The appropriate spline boundary conditions are derived for all procedures. For central finite differences, this leads to a two-point, second-order accurate generalization of the commonly used three-point end-difference formula. Solutions with several spline and Hermite procedures are presented for the boundary layer equations, with and without mass transfer, and for the incompressible viscous flow in a driven cavity. Divergence and nondivergence equations are considered for the cavity. Among the fourth-order techniques, it is shown that spline 4 has the smallest truncation error. The spline 4 procedure generally requires one-quarter the number of mesh points in a given coordinate direction as a central finite-difference calculation of equal accuracy. The Hermite 6 procedure leads to remarkably accurate boundary layer solutions.
Rubin, S. G.↗
The Construction of Curves and Surfaces Using Numerical Optimization Techniques
Numerical optimization techniques are playing an increasing role in curve and surface construction. Often difficult problems in curve and surface construction, especially when some aspect of shape control is involved, can be phrased as a constrained optimization problem. Four such classes of problems are explored: parametric curve fitting with non-linear shape constraints; explicit surface fitting with linear shape constraints; surface fitting to scattered data giving rise to ill-posed problems; finally, variable knot problems. In each of these problems there is a nonlinear aspect: either the shape of the curve or surface is important for manufacturing or engineering reasons or the shape affects the convergence of numerical algorithms which use the curve or surface or the placement of knots affects the accuracy of the fits. In all cases the class of functions used is that of parametric spline curves and tensor or direct product spline surfaces. The reason for choosing this class is that splines provide flexible models that are easily evaluated and stored. Furthermore, the B-spline representation of splines leads to convenient expressions for shape control over regions.
Ferguson, D. R.↗
Quantitative analysis of the reconstruction performance of interpolants
The analysis presented provides a quantitative measure of the reconstruction or interpolation performance of linear, shift-invariant interpolants. The performance criterion is the mean square error of the difference between the sampled and reconstructed functions. The analysis is applicable to reconstruction algorithms used in image processing and to many types of splines used in numerical analysis and computer graphics. When formulated in the frequency domain, the mean square error clearly separates the contribution of the interpolation method from the contribution of the sampled data. The equations provide a rational basis for selecting an optimal interpolant; that is, one which minimizes the mean square error. The analysis has been applied to a selection of frequently used data splines and reconstruction algorithms: parametric cubic and quintic Hermite splines, exponential and nu splines (including the special case of the cubic spline), parametric cubic convolution, Keys' fourth-order cubic, and a cubic with a discontinuous first derivative. The emphasis in this paper is on the image-dependent case in which no a priori knowledge of the frequency spectrum of the sampled function is assumed.
Lansing, Donald L.↗
Direct numerical simulation of incompressible axisymmetric flows
In the present work, we propose to conduct direct numerical simulations (DNS) of incompressible turbulent axisymmetric jets and wakes. The objectives of the study are to understand the fundamental behavior of axisymmetric jets and wakes, which are perhaps the most technologically relevant free shear flows (e.g. combuster injectors, propulsion jet). Among the data to be generated are various statistical quantities of importance in turbulence modeling, like the mean velocity, turbulent stresses, and all the terms in the Reynolds-stress balance equations. In addition, we will be interested in the evolution of large-scale structures that are common in free shear flow. The axisymmetric jet or wake is also a good problem in which to try the newly developed b-spline numerical method. Using b-splines as interpolating functions in the non-periodic direction offers many advantages. B-splines have local support, which leads to sparse matrices that can be efficiently stored and solved. Also, they offer spectral-like accuracy that are C(exp O-1) continuous, where O is the order of the spline used; this means that derivatives of the velocity such as the vorticity are smoothly and accurately represented. For purposes of validation against existing results, the present code will also be able to simulate internal flows (ones that require a no-slip boundary condition). Implementation of no-slip boundary condition is trivial in the context of the b-splines.
Loulou, Patrick↗
Student Support for Research in Hierarchical Control and Trajectory Planning
Generally, classical polynomial splines tend to exhibit unwanted undulations. In this work, we discuss a technique, based on control principles, for eliminating these undulations and increasing the smoothness properties of the spline interpolants. We give a generalization of the classical polynomial splines and show that this generalization is, in fact, a family of splines that covers the broad spectrum of polynomial, trigonometric and exponential splines. A particular element in this family is determined by the appropriate control data. It is shown that this technique is easy to implement. Several numerical and curve-fitting examples are given to illustrate the advantages of this technique over the classical approach. Finally, we discuss the convergence properties of the interpolant.
Martin, Clyde F.↗
Galerkin formulations of isogeometric shell analysis: Alleviating locking with Greville quadratures and higher-order elements
Here we propose new quadrature schemes that asymptotically require only four in-plane points for Reissner–Mindlin shell elements and nine in-plane points for Kirchhoff–Love shell elements in B-spline and NURBS-based isogeometric shell analysis, independent of the polynomial degree p of the elements. The quadrature points are Greville abscissae associated with pth-order B-spline basis functions whose continuities depend on the specific Galerkin formulations, and the quadrature weights are calculated by solving a linear moment fitting problem in each parametric direction. The proposed shell element formulations are shown through numerical studies to be rank sufficient and to be free of spurious modes. The studies reveal comparable accuracy, in terms of both displacement and stress, compared with fully integrated spline-based shell elements, while at the same time reducing storage and computational cost associated with forming element stiffness and mass matrices and force vectors. The high accuracy with low computational cost makes the proposed quadratures along with higher-order spline bases, in particular polynomial orders, p = 5 and 6, good choices for alleviating membrane and shear locking in shells.
42 ENGINEERING↗
SineKAN: Kolmogorov-Arnold Networks using sinusoidal activation functions
Recent work has established an alternative to traditional multi-layer perceptron neural networks in the form of Kolmogorov-Arnold Networks (KAN). The general KAN framework uses learnable activation functions on the edges of the computational graph followed by summation on nodes. The learnable edge activation functions in the original implementation are basis spline functions (B-Spline). Here, we present a model in which learnable grids of B-Spline activation functions are replaced by grids of re-weighted sine functions (SineKAN). We evaluate numerical performance of our model on a benchmark vision task. We show that our model can perform better than or comparable to B-Spline KAN models and an alternative KAN implementation based on periodic cosine and sine functions representing a Fourier Series. Further, we show that SineKAN has numerical accuracy that could scale comparably to dense neural networks (DNNs). Compared to the two baseline KAN models, SineKAN achieves a substantial speed increase at all hidden layer sizes, batch sizes, and depths. Current advantage of DNNs due to hardware and software optimizations are discussed along with theoretical scaling. Additionally, properties of SineKAN compared to other KAN implementations and current limitations are also discussed.
Reinhardt, Eric↗
Results of full-scale vortex attenuation flight experiments
Flight tests have been conducted to evaluate the effectiveness of a wing-tip vortex attenuating device, referred to as a spline. Vortex penetrations were made with a PA-28 behind a C-54 aircraft with and without wing-tip splines attached, and the resultant rolling acceleration was measured and related to the roll acceleration capability of the PA-28. Tests were conducted over a range of separation distances from about 5 n mi to less than 1 n mi. Preliminary results indicate that, with the splines installed, there was a significant reduction in the vortex-induced roll acceleration experienced by the PA-28 probe aircraft, and the distance at which the PA-28 roll control became ineffective was reduced from 2.5 n mi to less than 1 n mi. Although splines significantly reduced the C-54 rate of climb, the rates available with four engines were acceptable for this test program.
Hastings, E. C., Jr.↗
The separated turbulent boundary layer over a wavy wall
A study and application of the fourth order spline collocation procedure, numerical solution of boundary layer like differential equations, is presented. A simple inversion algorithm for the simultaneous solution of the resulting difference equations is given. Particular attention is focused on the boundary condition representation for the spline second derivative approximations. Solutions using the spline procedure, as well as the three point finite difference method, are presented for several model problems in order to assess and improve the spline numerical scheme. Application of the resulting algorithm to the incompressible laminar self similar boundary layer equations is presented.
Polak, A.↗
Time Varying Compensator Design for Reconfigurable Structures Using Non-Collocated Feedback
Analysis and synthesis tools are developed to improved the dynamic performance of reconfigurable nonminimum phase, nonstrictly positive real-time variant systems. A novel Spline Varying Optimal (SVO) controller is developed for the kinematic nonlinear system. There are several advantages to using the SVO controller, in which the spline function approximates the system model, observer, and controller gain. They are: The spline function approximation is simply connected, thus the SVO controller is more continuous than traditional gain scheduled controllers when implemented on a time varying plant; ft is easier for real-time implementations in storage and computational effort; where system identification is required, the spline function requires fewer experiments, namely four experiments; and initial startup estimator transients are eliminated. The SVO compensator was evaluated on a high fidelity simulation of the Shuttle Remote Manipulator System. The SVO controller demonstrated significant improvement over the present arm performance: (1) Damping level was improved by a factor of 3; and (2) Peak joint torque was reduced by a factor of 2 following Shuttle thruster firings.
Scott, Michael A.↗
Total Blood Mercury Predicts Methylmercury Exposure in Fish and Shellfish Consumers
Abstract Many studies evaluating methylmercury (MeHg) toxicity rely on whole blood total mercury (THg) measurements to estimate MeHg exposure. However, whole blood THg includes other forms of mercury (Hg), such as inorganic Hg, which have different exposure sources and toxicological effects than MeHg. Therefore, estimating the whole blood MeHg/THg ratio is critical to predicting MeHg exposure and, subsequently, efforts to establish an exposure–response relationship for use in risk assessment. A large, representative dataset (National Health and Nutrition Examination Survey (NHANES) 2011–2016) was used to determine the whole blood MeHg/THg ratio among (a) self-reported fish and shellfish consumers, ≥ 15 years of age (the “full adult” population ( N = 5268 training dataset; N = 2336 test dataset)) and (b) female fish and shellfish consumers, 15–44 years of age (the “women of reproductive age” population ( N = 1285 training dataset; N = 560 test dataset)). Unadjusted and adjusted linear and spline models with direct measurements for both THg and MeHg were evaluated. The mean whole blood MeHg/THg ratio was 0.75 (95% confidence interval (CI): 0.74, 0.75). This ratio was significantly higher among those with higher THg concentrations. All models exhibited excellent fit (adjusted R 2 from 0.957 to 0.982). Performance was slightly improved in spline versus linear models. For the full adult population and women of reproductive age, the unadjusted spline model predicted whole blood MeHg concentrations of 5.65 µg/L and 5.55 µg/L, respectively, when the THg concentration was 5.80 µg/L. These results suggest that whole blood THg is a good predictor of whole blood MeHg among fish and shellfish consumers.
Wells, Ellen M.↗