Derek C. Thomas

Hierarchical T-splines: Analysis-suitability, Bézier extraction, and application as an adaptive basis for isogeometric analysis

Abstract In this paper hierarchical analysis-suitable T-splines (HASTS) are developed. The resulting spaces are a superset of both analysis-suitable T-splines and hierarchical B-splines. The additional flexibility provided by the hierarchy of T-spline spaces results in simple, highly localized refinement algorithms which can be utilized in a design or analysis context. A detailed theoretical formulation is presented. Bezier extraction is extended to HASTS simplifying the implementation of HASTS in existing finite element codes. The behavior of a simple HASTS refinement algorithm is compared to the local refinement algorithm for analysis-suitable T-splines demonstrating the superior efficiency and locality of the HASTS algorithm. Finally, HASTS are utilized as a basis for adaptive isogeometric analysis.

Bézier projection: A unified approach for local projection and quadrature-free refinement and coarsening of NURBS and T-splines with particular application to isogeometric design and analysis

Abstract We introduce Bezier projection as an element-based local projection methodology for B-splines, NURBS, and T-splines. This new approach relies on the concept of Bezier extraction and an associated operation introduced here, spline reconstruction, enabling the use of Bezier projection in standard finite element codes. Bezier projection exhibits provably optimal convergence and yields projections that are virtually indistinguishable from global L 2 projection. Bezier projection is used to develop a unified framework for spline operations including cell subdivision and merging, degree elevation and reduction, basis roughening and smoothing, and spline reparameterization. In fact, Bezier projection provides a quadrature-free approach to refinement and coarsening of splines. In this sense, Bezier projection provides the fundamental building block for h p k r -adaptivity in isogeometric analysis.

Evolution of the average steepening factor for nonlinearly propagating waves

Difficulties arise in attempting to discern the effects of nonlinearity in near-field jet-noise measurements due to the complicated source structure of high-velocity jets. This article describes a measure that may be used to help quantify the effects of nonlinearity on waveform propagation. This measure, called the average steepening factor (ASF), is the ratio of the average positive slope in a time waveform to the average negative slope. The ASF is the inverse of the wave steepening factor defined originally by Gallagher [AIAA Paper No. 82-0416 (1982)]. An analytical description of the ASF evolution is given for benchmark cases-initially sinusoidal plane waves propagating through lossless and thermoviscous media. The effects of finite sampling rates and measurement noise on ASF estimation from measured waveforms are discussed. The evolution of initially broadband Gaussian noise and signals propagating in media with realistic absorption are described using numerical and experimental methods. The ASF is found to be relatively sensitive to measurement noise but is a relatively robust measure for limited sampling rates. The ASF is found to increase more slowly for initially Gaussian noise signals than for initially sinusoidal signals of the same level, indicating the average distortion within noise waveforms occur more slowly.

Evolution of the derivative skewness for nonlinearly propagating waves

The skewness of the first time derivative of a pressure waveform, or derivative skewness, has been used previously to describe the presence of shock-like content in jet and rocket noise. Despite its use, a quantitative understanding of derivative skewness values has been lacking. In this paper, the derivative skewness for nonlinearly propagating waves is investigated using analytical, numerical, and experimental methods. Analytical expressions for the derivative skewness of an initially sinusoidal plane wave are developed and, along with numerical data, are used to describe its behavior in the preshock, sawtooth, and old-age regions. Analyses of common measurement issues show that the derivative skewness is relatively sensitive to the effects of a smaller sampling rate, but less sensitive to the presence of additive noise. In addition, the derivative skewness of nonlinearly propagating noise is found to reach greater values over a shorter length scale relative to sinusoidal signals. A minimum sampling rate is recommended for sinusoidal signals to accurately estimate derivative skewness values up to five, which serves as an approximate threshold indicating significant shock formation.

Phase and amplitude gradient method for the estimation of acoustic vector quantities

An alternative pressure-sensor based method for estimating the acoustic intensity, the phase and amplitude gradient estimation (PAGE) method, is presented. This method uses the same hardware as the standard finite-difference method, but does not suffer from the frequency-dependent bias inherent to the finite-difference method. A detailed derivation of the PAGE method and the finite-difference method is presented. Both methods are then compared using simple acoustic fields. The ability to unwrap the phase component of the PAGE method is discussed, which leads to accurate intensity estimates above previous frequency limits. The uncertainties associated with both methods of estimation are presented. It is shown that the PAGE method provides more accurate intensity estimates over a larger frequency bandwidth.

A balloon lens: Acoustic scattering from a penetrable sphere

A balloon filled with a gas that has a different sound speed than that of air has been used as an acoustic lens. One purpose of the lens is to show refraction of sound waves in an analogy to geometric optics. We discuss the physics of the balloon lens demonstration. To determine the validity of a gas-filled balloon as a classroom demonstration of an acoustic lens and to understand the corresponding phenomena, its physics is considered analytically, numerically, and experimentally. Our results show that although a geometric analogy is a good first-order approximation, scattering theory is required to fully understand the observed phenomena. Thus this demonstration can be adapted to a wide range of students, from those learning the basic principles of refraction to advanced students studying scattering.

Comparison of Two Time-domain Measures of Nonlinearity in Near-field Propagation of High-power Jet Noise

Time-domain metrics are used to investigate the nonlinearity of the sound in the vicinity of the F-35 AA-1. The first measure considered is the average steepening factor (ASF), which we define as the inverse of the wave steepening factor and is a ratio of the expectation value of the positive slopes in the waveform to the expectation value of the negative slopes. The second nonlinearity metric is the skewness of the time derivative of the pressure waveform (derivative skewness), which describes the asymmetry of the distribution of slopes in the waveform. Spatial maps of both metrics applied to the F-35 AA-1 data reveal that regions of increasing derivative skewness correspond more closely to the maximum sound radiation area, whereas the largest values for the ASF seem aligned with the regions where the waveform amplitude distributions are most asymmetric. It is proposed that these two metrics reveal different characteristics of the nonlinear propagation of jet noise. The ASF is more representative of the average slopes, which are dominated by high frequencies. Conversely, the derivative skewness identifies of large positive slopes and hence relates to the shock content in the noise.

Isogeometric analysis based on T-splines

This chapter provides an introduction to the use of T-splines in isogeometric analysis. A simple definition of two-dimensional T-splines is given and Bezier extraction is introduced. The basic details for implementation of T-splines as finite element shape functions are given. Two examples of integrated analysis and design based on commercial tools are given to illustrate the utility of T-spline-based IGA in a design-through-analysis workflow.

Time-domain effects of rigid sphere scattering on measurement of transient plane waves

Transient waves, like all other acoustic waves, will diffract around solid objects, such as measurement instrumentation. A derivation of an impulse response function on the surface of a rigid sphere, based on linear, classical scattering theory, is presented. The theoretical impulse response function is validated using an experiment with blast noise. An application of the impulse response function to a rocket noise measurement is discussed. The impulse response function shows that the presence of the rigid sphere significantly affects the measurement and estimation of rocket-noise waveforms, power spectral densities, and statistical measures.

Bézier B̄ projection

Abstract In this paper we demonstrate the use of Bezier projection to alleviate locking phenomena in structural mechanics applications of isogeometric analysis. Interpreting the well-known B projection in two different ways we develop two formulations for locking problems in beams and nearly incompressible elastic solids. One formulation leads to a sparse symmetric system and the other leads to a sparse non-symmetric system. To demonstrate the utility of Bezier projection for both geometry and material locking phenomena we focus on transverse shear locking in Timoshenko beams and volumetric locking in nearly compressible linear elasticity although the approach can be applied generally to other types of locking phenomena as well. Bezier projection is a local projection technique with optimal approximation properties, which in many cases produces solutions that are comparable to global L 2 projection. In the context of B methods, the use of Bezier projection produces sparse stiffness matrices with only a slight increase in bandwidth when compared to standard displacement-based methods. Of particular importance is that the approach is applicable to any spline representation that can be written in Bezier form like NURBS, T-splines, LR-splines, etc. We discuss in detail how to integrate this approach into an existing finite element framework with minimal disruption through the use of Bezier extraction operators and a newly introduced dual basis for the Bezier projection operator. We then demonstrate the behavior of the two proposed formulations through several challenging benchmark problems.