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</math>
where <math>k</math> is the [[wavenumber]], <math>\mathbf{I}</math> is the [[identity matrix]], and <math>\mathbf{r}</math> is the vector from the source dipole to the observation point. Evaluating the derivatives leads to the explicit form:
:<math>
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</math>
where <math>\hat{\mathbf{r}} = \mathbf{r} / |\mathbf{r}|</math> is the [[unit vector]] pointing from the source to the observation point.
This Green’s tensor describes the electric field generated by a dipole in a homogeneous medium. It is used to compute the off-diagonal blocks of the interaction matrix in DDA, that is, the interaction between distinct dipoles <math>j \ne k</math>. The singular self-term <math>\mathbf{G}(\mathbf{r} = 0)</math> is excluded and replaced by a prescribed local term involving the inverse polarizability tensor <math>\boldsymbol{\alpha}_j^{-1}</math>.
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==Fast Fourier Transform for fast convolution calculations==
The use of the fast Fourier transform (FFT) to accelerate convolution operations in the discrete dipole approximation (DDA) was introduced by Goodman, Draine, and Flatau in 1991{{r|Goodman1991}}. Their method employed the three-dimensional FFT algorithm (GPFA) developed by Clive Temperton{{r|
A similar variant to that of Goodman, Draine, and Flatau was adopted in the 2021 MATLAB implementation by Shabaninezhad and Ramakrishna{{r|matlab2021}}. In this approach, the computational ___domain for the polarization vector is zero-padded to <math>(2n_x-1)\times (2n_y-1)\times (2n_z-1)</math> instead of the <math>2n_x\times 2n_y\times 2n_z</math> sizes used in ''DDSCAT''. The stored interaction matrix <math>G</math> differs from ''DDSCAT'' in that there is no zero plane inserted between the positive and negative offsets along each axis. The FFT is performed as a sequence of one-dimensional transforms along the <math>x</math>, <math>y</math>, and <math>z</math> axes, which is mathematically equivalent to performing a full 3-D FFT on the padded ___domain.
Sequence of 1D FFTs was used by MacDonald in his Ph. D. thesis <ref name=mcdonald2009/>.
Barrowes method is a numerical technique for multiplying an <math>n</math>-dimensional block Toeplitz matrix by a vector using the fast Fourier transform (FFT). In three dimensions with grid sizes <math>n_x</math>, <math>n_y</math>, and <math>n_z</math>, the method embeds the block Toeplitz array into a larger block circulant array of size <math>(2n_x-1)\times (2n_y-1)\times (2n_z-1)</math>, which ensures that the corresponding convolution is free of cyclic wraparound. The kernel—containing all positive and negative offsets with the self term set to zero—is reversed along the offset axes, flattened into a one-dimensional array, and transformed by a single long FFT. The input vector is likewise placed in a zero-padded ___domain of the same size, flattened, and transformed. An element-wise product in the frequency ___domain corresponds to the spatial-___domain convolution; an inverse FFT is then reshaped and cropped back to the physical ___domain to obtain the result. The method applies to arbitrary dimension <math>n</math> and block size, and was used originally in the discrete dipole approximation.<ref name="Barrowes2001" />
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<ref name=chaumet2022discrete>{{cite journal |last=Chaumet |first=Patrick C. |title=The discrete dipole approximation: A review |journal=Mathematics |volume=10 |issue=17 |pages=3049 |year=2022 |doi=10.3390/math10173049 |doi-access=free }}</ref>
<ref name=Barrowes2001>{{cite journal |last1=Barrowes |first1=B. E. |last2=Teixeira |first2=F. L. |last3=Kong |first3=J. A. |title=Fast algorithm for matrix–vector multiply of asymmetric multilevel block-Toeplitz matrices in 3-D scattering |journal=Microw. Opt. Technol. Lett. |volume=31 |issue=1 |pages=28–32 |year=2001 |doi=10.1002/mop.1348}}</ref>
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