Spherical harmonic analysis and synthesis for global multiresolution applications

Blais, J. A. R.; Provins, D. A.

doi:10.1007/s001900100217

Cited by 26 publications

(14 citation statements)

References 11 publications

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“…Taking our problem for example, the desired degree of precision for V lm is 4, which means the degrees of both v(θ, φ) and Y * lm (θ, φ) need to be at least 4, so 2N − 1 must be no less than 2 × 4 = 8; and because N has to be a power of 2, the smallest N satisfying these requirements is 8, which means we need 16 × 16 discrete data to ensure the accuracy. The Gaussian-Lengendre quadrature, on the other hand, is known for the ability to get to (2N − 1)th degree of accuracy with 2N × N discrete data [48][49][50][51]. The numerical integration is written:…”

Section: Discretisation Of the Problemmentioning

confidence: 99%

A spherical harmonic approach for the determination of HCP texture from ultrasound: A solution to the inverse problem

Lan

Lowe

Dunne

2015

Journal of the Mechanics and Physics of Solids

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Section: Discretisation Of the Problemmentioning

confidence: 99%

A spherical harmonic approach for the determination of HCP texture from ultrasound: A solution to the inverse problem

Lan

Lowe

Dunne

2015

Journal of the Mechanics and Physics of Solids

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“…The latter can be achieved by means of numerical integrations, and the choices of positions to obtain the discrete data are called quadratures. Comparisons of the two most widely used quadrature, namely equiangular and Gaussian-Legendre, were presented in the literature [18,27,28], and in this paper we continue to choose the latter for its ability to get to (2N − 1)th degree of accuracy with only 2N × N discrete data. The numerical integration is written:…”

Section: 1mentioning

confidence: 99%

A generalized spherical harmonic deconvolution to obtain texture of cubic materials from ultrasonic wave speed

Lan

Lowe

Dunne

2015

Journal of the Mechanics and Physics of Solids

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In this paper, the spherical harmonic convolution approach for HCP materials [1] is extended into a generalised form for the principal purpose of bulk texture determination in cubic polycrystals from ultrasonic wave speed measurements. It is demonstrated that the wave speed function of a general single crystal convolves with the polycrystal Orientation Distribution Function (ODF) to make the resultant polycrystal wave speed function such that when the three functions are expressed in harmonic expansions, the coefficients of any one function may be determined from the coefficients of the other two. All three Euler angles are taken in account in the description of the ODF such that the theorem applies for all general crystal systems.The forward problem of predicting polycrystal wave speed with knowledge of single crystal properties and the ODF is solved for all general cases, with validation carried out on cubic textures showing strong sensitivity to texture and excellent quantitative accuracy in predicted wave speed amplitudes. Importantly, it is also revealed by the theorem that the cubic structure is one of only two crystal systems (the other being HCP) whose orientation distributions can be inversely determined from polycrystal wave velocities by virtue of their respective crystal symmetries. Proof of principle is then established by recovering the ODFs of representative cubic textures solely from the wave velocities generated from a computational model using these texture inputs, and excellent accuracies are achieved in the recovered ODF coefficients as well as the resultant pole figures. Hence the methodology is argued to provide a powerful technique for wave propagation studies and bulk texture measurement in cubic polycrystals and beyond.

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“…[11] and [3] for details). The tilde "~" will be used to indicate geodetic normalization in the following.…”

Section: Continuous and Discrete Shtsmentioning

confidence: 99%

“…Furthermore, there are several strategies with recursions in n and m and these are far from being numerically equivalent (see e.g. [1,3,14]). The geodetically normalized associated Legendre functions nm P (cos ) θ are computed as a lower triangular matrix with the rows corresponding to the degrees n and the columns corresponding to the orders m. With the initialization for degrees and orders 0 and 1, 00 10 P (cos ) 1, P (cos ) 3 cos θ = θ = θ and 11 P (cos ) 3sin , θ = θ the diagonal terms nn n 1,n 1 P (cos ) (2n 1) / 2n sin P (cos ) [18], based on [10,15].…”

Section: Numerical Preconditioning and Optimizationmentioning

confidence: 99%

Discrete Spherical Harmonic Transforms: Numerical Preconditioning and Optimization

Blais

2008

Computational Science – ICCS 2008

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Abstract. Spherical Harmonic Transforms (SHTs) which are essentially Fourier transforms on the sphere are critical in global geopotential and related applications. Among the best known strategies for discrete SHTs are Chebychev quadratures and least squares. The numerical evaluation of the Legendre functions are especially challenging for very high degrees and orders which are required for advanced geocomputations. The computational aspects of SHTs and their inverses using both quadrature and least-squares estimation methods are discussed with special emphasis on numerical preconditioning that guarantees reliable results for degrees and orders up to 3800 in REAL*8 or double precision arithmetic. These numerical results of spherical harmonic synthesis and analysis using simulated spectral coefficients are new and especially important for a number of geodetic, geophysical and related applications with ground resolutions approaching 5 km.

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Spherical harmonic analysis and synthesis for global multiresolution applications

Cited by 26 publications

References 11 publications

A spherical harmonic approach for the determination of HCP texture from ultrasound: A solution to the inverse problem

A spherical harmonic approach for the determination of HCP texture from ultrasound: A solution to the inverse problem

A generalized spherical harmonic deconvolution to obtain texture of cubic materials from ultrasonic wave speed

Discrete Spherical Harmonic Transforms: Numerical Preconditioning and Optimization

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