On the Harmonic Balance Method Augmented with Non-Smooth Basis Functions for Contact/Impact Problems.
Abstract not provided.
Saunders, Brian E↗
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Design of Multi-Parameter Steerable Functions Using Cascade Basis Reduction
A new cascade basis reduction method of computing the optimal least-squares set of basis functions steering a given function is presented. The method combines the Lie group-theoretic and the singular value decomposition approaches in such a way that their respective strengths complement each other. Since the Lie group-theoretic approach is used, the set of basis and steering functions computed can be expressed analytically. Because the singular value decomposition method is used, this set of basis and steering functions is optimal in the least-squares sense. Furthermore, the computational complexity in designing basis functions for transformation groups with large numbers of parameters is significantly reduced. The efficiency of the cascade basis reduction method is demonstrated by designing a set of basis functions that steers a Gabor function under the four-parameter linear transformation group.
Teo, P.↗
A flexible class of priors for orthonormal matrices with basis function-specific structure
Statistical modeling of high-dimensional matrix-valued data motivates the use of a low-rank representation that simultaneously summarizes key characteristics of the data and enables dimension reduction. Low-rank representations commonly factor the original data into the product of orthonormal basis functions and weights, where each basis function represents an independent feature of the data. However, the basis functions in these factorizations are typically computed using algorithmic methods that cannot quantify uncertainty or account for basis function correlation structure a priori. While there exist Bayesian methods that allow for a common correlation structure across basis functions, empirical examples motivate the need for basis function-specific dependence structure. We propose a prior distribution for orthonormal matrices that can explicitly model basis function-specific structure. The prior is used within a general probabilistic model for singular value decomposition to conduct posterior inference on the basis functions while accounting for measurement error and fixed effects. We discuss how the prior specification can be used for various scenarios and demonstrate favorable model properties through synthetic data examples. Finally, we apply our method to two-meter air temperature data from the Pacific Northwest, enhancing our understanding of the Earth system’s internal variability.
97 MATHEMATICS AND COMPUTING↗
Dynamical model for longitudinal wave functions in light-front holographic QCD
We construct a Schrödinger-like equation for the longitudinal wave function of a meson in the valence qq{sup -bar} sector, based on the ’t Hooft model for large-N two-dimensional QCD, and combine this with the usual transverse equation from light-front holographic QCD, to obtain a model for mesons with massive quarks. The computed wave functions are compared with the wave function ansatz of Brodsky and de Téramond and used to compute decay constants and parton distribution functions. The basis functions used to solve the longitudinal equation may be useful for more general calculations of meson states in QCD. -- Highlights: •Provide relativistic quark model based on light-front holographic QCD. •Incorporate dependence on quark mass. •Consistent with the Brodsky–de Téramond quark-wave-function ansatz. •Compute meson decay constants and parton distribution functions. •Illustrate use of basis functions that could be convenient for more general numerical calculations in light-front QCD.
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
A numerical study of viscous vortex rings using a spectral method
Viscous, axisymmetric vortex rings are investigated numerically by solving the incompressible Navier-Stokes equations using a spectral method designed for this type of flow. The results presented are axisymmetric, but the method is developed to be naturally extended to three dimensions. The spectral method relies on divergence-free basis functions. The basis functions are formed in spherical coordinates using Vector Spherical Harmonics in the angular directions, and Jacobi polynomials together with a mapping in the radial direction. Simulations are performed of a single ring over a wide range of Reynolds numbers (Re approximately equal gamma/nu), 0.001 less than or equal to 1000, and of two interacting rings. At large times, regardless of the early history of the vortex ring, it is observed that the flow approaches a Stokes solution that depends only on the total hydrodynamic impulse, which is conserved for all time. At small times, from an infinitely thin ring, the propagation speeds of vortex rings of varying Re are computed and comparisons are made with the asymptotic theory by Saffman. The results are in agreement with the theory; furthermore, the error is found to be smaller than Saffman's own estimate by a factor square root ((nu x t)/R squared) (at least for Re=0). The error also decreases with increasing Re at fixed core-to-ring radius ratio, and appears to be independent of Re as Re approaches infinity). Following a single ring, with Re=500, the vorticity contours indicate shedding of vorticity into the wake and a settling of an initially circular core to a more elliptical shape, similar to Norbury's steady inviscid vortices. Finally, we consider the case of leapfrogging vortex rings with Re=1000. The results show severe straining of the inner vortex core in the first pass and merging of the two cores during the second pass.
Stanaway, S. K.↗
UQ Toolkit v 2.0
The Uncertainty Quantification (UQ) Toolkit is a software library for the characterizaton and propagation of uncertainties in computational models. For the characterization of uncertainties, Bayesian inference tools are provided to infer uncertain model parameters, as well as Bayesian compressive sensing methods for discovering sparse representations of high-dimensional input-output response surfaces, and also Karhunen-Loève expansions for representing stochastic processes. Uncertain parameters are treated as random variables and represented with Polynomial Chaos expansions (PCEs). The library implements several spectral basis function types (e.g. Hermite basis functions in terms of Gaussian random variables or Legendre basis functions in terms of uniform random variables) that can be used to represent random variables with PCEs. For propagation of uncertainty, tools are provided to propagate PCEs that describe the input uncertainty through the computational model using either intrusive methods (Galerkin projection of equations onto basis functions) or non-intrusive methods (perform deterministic operation at sampled values of the random values and project the obtained results onto basis functions).
Safta, Cosmin↗
UQ Toolkit v 2.0
The Uncertainty Quantification (UQ) Toolkit is a software library for the characterizaton and propagation of uncertainties in computational models. For the characterization of uncertainties, Bayesian inference tools are provided to infer uncertain model parameters, as well as Bayesian compressive sensing methods for discovering sparse representations of high-dimensional input-output response surfaces, and also Karhunen-Loève expansions for representing stochastic processes. Uncertain parameters are treated as random variables and represented with Polynomial Chaos expansions (PCEs). The library implements several spectral basis function types (e.g. Hermite basis functions in terms of Gaussian random variables or Legendre basis functions in terms of uniform random variables) that can be used to represent random variables with PCEs. For propagation of uncertainty, tools are provided to propagate PCEs that describe the input uncertainty through the computational model using either intrusive methods (Galerkin projection of equations onto basis functions) or non-intrusive methods (perform deterministic operation at sampled values of the random values and project the obtained results onto basis functions).
Source record↗
UQ Toolkit v. 3.0
The Uncertainty Quantification (UQ) Toolkit is a software library for the characterization and propagation of uncertainties in computational models. This library provides Bayesian inference tools to infer uncertain model parameters, Bayesian compressive sensing methods for discovering sparse representations of high-dimensional input-output response surfaces, methods for constructing Karhunen-Loève representations of stochastic processes, and global sensitivity analysis tools used to compute Sobol indices in order to characterize the importance of uncertain inputs or the interactions between them. The basis for many of these methods relies on representing random variables with Polynomial Chaos expansions (PCEs). This library implements several spectral basis function types (e.g. Hermite basis functions in terms of Gaussian random variables or Legendre basis functions in terms of uniform random variables) that can be used to represent random variables with PCEs. For the propagation of uncertainty, tools are provided to propagate PCEs that describe the input uncertainty through the computational model using either intrusive methods (Galerkin projection of equations onto basis functions) or non-intrusive methods (perform deterministic operation at sampled values of the random values and project the obtained results onto basis functions).
Sargsyan, Khachik↗
Fully-Implicit Orthogonal Reconstructed Discontinuous Petrov-Galerkin Method for Multiphysics Problems
This document is focussed on the development of reconstructed discontinuous Galerkin (RDG P n P m ) method, based on orthogonal basis functions. Orthogonality of basis functions is essential for enabling robust and efficient fully-implicit Newton-Krylov based time discretization. The method is designed for generic partial differential equations, including transient hyperbolic, parabolic and/or elliptic operators, which are attributed to many multiphysics problems. Our driving application of interest is additive manufacturing, which requires simulations of fluid-solid-gas systems, with compressibility effects and phase change (melting/ solidification). Thus, we demonstrate the method capabilities for solving nonlinear heat conduction equation coupled with multi-material Navier-Stokes based compressible fluid (in low Mach number limit) systems. We focus our attention to the method accuracy (in both time and space), as well as solvability of linear algebra involved in linear steps of the Newton-based solver. We demonstrate the accuracy and efficacy of the method, emphasizing the advantages from the orthogonality of basis functions, which makes better conditioning of underlying (approximate) Jacobian matrices, and rapid convergence of Krylov methods. iii
97 MATHEMATICS AND COMPUTING↗
Neural chaos: A spectral stochastic neural operator
Building surrogate models for operators with uncertainty quantification capabilities is essential for many engineering applications where randomness–such as variability in material properties, boundary conditions, and initial conditions–is unavoidable. Polynomial Chaos Expansion (PCE) is widely recognized as a go-to method for constructing stochastic surrogates in both intrusive and non-intrusive ways, and it has recently been used in the context of operator learning. However, its application becomes challenging for complex or high-dimensional processes, as achieving accuracy requires higher-order polynomials, which can increase computational demand and/or the risk of overfitting. Furthermore, PCE requires specialized treatments to manage random variables that are not independent, and these treatments may be problem-dependent or may fail with increasing complexity. Here, in this work, we adopt the same formalism as the spectral expansion used in PCE; however, we replace the classical polynomial basis functions with neural network (NN) basis functions to leverage their expressivity. To achieve this, we propose an algorithm that identifies NN-parameterized basis functions in a purely data-driven manner, without any prior assumptions about the joint distribution of the random variables involved, whether independent or dependent, or about their marginal distributions. The proposed algorithm identifies each NN-parameterized basis function sequentially, ensuring they are orthogonal with respect to the data distribution. The basis functions are constructed directly on the joint stochastic variables without requiring a tensor product structure or assuming independence of the random variables. This approach may offer greater flexibility for complex stochastic models, while simplifying implementation compared to the tensor product structures typically used in PCE to handle random vectors. This is particularly advantageous given the current state of open-source packages, where building and training neural networks can be done with just a few lines of code and extensive community support. We demonstrate the effectiveness of the proposed scheme through several numerical examples of varying complexity and provide comparisons with classical PCE.
Polynomial chaos expansion↗
Evolvable synthetic neural system
An evolvable synthetic neural system includes an evolvable neural interface operably coupled to at least one neural basis function. Each neural basis function includes an evolvable neural interface operably coupled to a heuristic neural system to perform high-level functions and an autonomic neural system to perform low-level functions. In some embodiments, the evolvable synthetic neural system is operably coupled to one or more evolvable synthetic neural systems in a hierarchy.
Curtis, Steven A.↗
A Characteristics Approach to the Finite Element Method
Herein, we present a new method for solving the linear Boltzmann transport equation. Two commonly used and well-understood methods for solving partial differential equations are the method of characteristics (MOC) and the finite element method (FEM). We propose a new method that combines the fundamental concept of the FEM with the analytic solution from the MOC to obtain coefficients for the FEM basis function expansion. Traditionally, coefficients for the FEM basis function expansion are obtained via matrix inversion. Instead, we solve for the coefficients with the MOC and represent the underlying fields with the basis function expansion using these coefficients. We provide a convergence study for our method with results from two sets of FEM basis functions: Gauss-Legendre and Gauss-Lobatto sets. We also compare two different variations of our method categorized as short characteristics and intermediate characteristics.
42 ENGINEERING↗
Linear and Geometrically Nonlinear Structural Shape Sensing from Strain Data
An aircraft’s shape must be measured during flight to implement active shape controls such as active trim shape, divergence, and sonic boom controls. A basis function method for linear as well as geometrically nonlinear structural shape sensing is proposed in this study. Basis functions can be mode shapes, linear static deformation shapes, and/or geometrically nonlinear static deformation shapes. The proposed basis function method is validated using a high-aspect-ratio wing, a swept test plate, and the National Aeronautics and Space Administration X-59 aircraft. A large structural deformation of a high-aspect-ratio wing is successfully captured using the proposed basis function method with less than 0.3%prediction error at the wing-tip section. The basis function method gives excellent deformation prediction, even with stress concentration. Performance of the basis functions is compared using the X-59 stabilator sample case. In most of the sample load cases, static deformations give a better correlation with the target deformation than do mode shapes. Wing shape sensing with sparse strain data are also demonstrated in this study using the X-59 aircraft. Predicted structural deformations match reasonably with the target deformations, even without strain gauges on some structural components. The predicted deformations have a good match with the target deformations.
Static Deformation↗
A 3D High-Order Unstructured Finite-Volume Algorithm for Solving Maxwell's Equations
A three-dimensional finite-volume algorithm based on arbitrary basis functions for time-dependent problems on general unstructured grids is developed. The method is applied to the time-domain Maxwell equations. Discrete unknowns are volume integrals or cell averages of the electric and magnetic field variables. Spatial terms are converted to surface integrals using the Gauss curl theorem. Polynomial basis functions are introduced in constructing local representations of the fields and evaluating the volume and surface integrals. Electric and magnetic fields are approximated by linear combinations of these basis functions. Unlike other unstructured formulations used in Computational Fluid Dynamics, the new formulation actually does not reconstruct the field variables at each time step. Instead, the spatial terms are calculated in terms of unknowns by precomputing weights at the beginning of the computation as functions of cell geometry and basis functions to retain efficiency. Since no assumption is made for cell geometry, this new formulation is suitable for arbitrarily defined grids, either smooth or unsmooth. However, to facilitate the volume and surface integrations, arbitrary polyhedral cells with polygonal faces are used in constructing grids. Both centered and upwind schemes are formulated. It is shown that conventional schemes (second order in Cartesian grids) are equivalent to the new schemes using first degree polynomials as the basis functions and the midpoint quadrature for the integrations. In the new formulation, higher orders of accuracy are achieved by using higher degree polynomial basis functions. Furthermore, all the surface and volume integrations are carried out exactly. Several model electromagnetic scattering problems are calculated and compared with analytical solutions. Examples are given for cases based on 0th to 3rd degree polynomial basis functions. In all calculations, a centered scheme is applied in the interior, while an upwind matching scheme is employed at material interfaces and the Engquist-Majda non-reflecting boundary condition is implemented at the numerical outer boundaries. The staggered leapfrog scheme and the Runge-Kutta methods are utilized for the time integration. Excellent agreements are found between the numerical and analytical solutions.
Liu, Yen↗
Generalized Multiscale Finite Element Methods with energy minimizing oversampling
In this work, we propose a general concept for constructing multiscale basis functions within Generalized Multiscale Finite Element Method, which uses oversampling and stable decomposition. The oversampling refers to using larger regions in constructing multiscale basis functions and stable decomposition allows estimating the local errors. The analysis of multiscale methods involves decomposing the error by coarse regions, where each error contribution is estimated. In this estimate, we often use oversampling techniques to achieve a fast convergence. We demonstrate our concepts in the mixed, the Interior Penalty Discontinuous Galerkin, and Hybridized Discontinuous Galerkin discretizations. One of the important features of the proposed basis functions is that they can be used in online Generalized Multiscale Finite Element Method, where one constructs multiscale basis functions using residuals. In these problems, it is important to achieve a fast convergence, which can be guaranteed if we have a stable decomposition. In our numerical results, we present examples for both offline and online multiscale basis functions. Our numerical results show that one can achieve a fast convergence when using online basis functions. Moreover, we observe that coupling using Hybridized Discontinuous Galerkin provides a better accuracy compared with Interior Penalty Discontinuous Galerkin, which is due to using multiscale glueing functions.
42 ENGINEERING↗
A resolution independent neural operator
The Deep operator network (DeepONet) is a powerful yet simple neural operator architecture that utilizes two deep neural networks to learn mappings between infinite-dimensional function spaces. This architecture is highly flexible, allowing the evaluation of the solution field at any location within the desired domain. However, it imposes a strict constraint on the input space, requiring all input functions to be discretized at the same locations; this limits its practical applications. Here, in this work, we introduce a general framework for operator learning from input–output data with arbitrary number and locations of sensors. This begins by introducing a resolution-independent DeepONet (RI-DeepONet), enabling it to handle input functions that are arbitrarily, but sufficiently finely, discretized. To this end, we propose two dictionary learning algorithms to adaptively learn a set of appropriate continuous basis functions, parameterized as implicit neural representations (INRs), from correlated signals defined on arbitrary point cloud data. These basis functions are then used to project arbitrary input function data as a point cloud onto an embedding space (i.e., a vector space of finite dimensions) with dimensionality equal to the dictionary size, which can be directly used by DeepONet without any architectural changes. In particular, we utilize sinusoidal representation networks (SIRENs) as trainable INR basis functions. The introduced dictionary learning algorithms are then used in a similar way to learn an appropriate dictionary of basis functions for the output function data, which defines a new neural operator architecture referred to as the R esolution I ndependent N eural O perator (RINO). In the RINO, the operator learning task simplifies to learning a mapping from the coefficients of input basis functions to the coefficients of output basis functions. We demonstrate the robustness and applicability of RINO in handling arbitrarily (but sufficiently richly) sampled input and output functions during both training and inference through several numerical examples.
Deep operator network (DeepONet)↗
An adaptive ANOVA-based PCKF for high-dimensional nonlinear inverse modeling
The probabilistic collocation-based Kalman filter (PCKF) is a recently developed approach for solving inverse problems. It resembles the ensemble Kalman filter (EnKF) in every aspect—except that it represents and propagates model uncertainty by polynomial chaos expansion (PCE) instead of an ensemble of model realizations. Previous studies have shown PCKF is a more efficient alternative to EnKF for many data assimilation problems. However, the accuracy and efficiency of PCKF depends on an appropriate truncation of the PCE series. Having more polynomial chaos basis functions in the expansion helps to capture uncertainty more accurately but increases computational cost. Selection of basis functions is particularly important for high-dimensional stochastic problems because the number of polynomial chaos basis functions required to represent model uncertainty grows dramatically as the number of input parameters (random dimensions) increases. In classic PCKF algorithms, the PCE basis functions are pre-set based on users' experience. Also, for sequential data assimilation problems, the basis functions kept in PCE expression remain unchanged in different Kalman filter loops, which could limit the accuracy and computational efficiency of classic PCKF algorithms. To address this issue, we present a new algorithm that adaptively selects PCE basis functions for different problems and automatically adjusts the number of basis functions in different Kalman filter loops. The algorithm is based on adaptive functional ANOVA (analysis of variance) decomposition, which approximates a high-dimensional function with the summation of a set of low-dimensional functions. Thus, instead of expanding the original model into PCE, we implement the PCE expansion on these low-dimensional functions, which is much less costly. We also propose a new adaptive criterion for ANOVA that is more suited for solving inverse problems. The new algorithm was tested with different examples and demonstrated great effectiveness in comparison with non-adaptive PCKF and EnKF algorithms.
97 MATHEMATICS AND COMPUTING↗