Continue to Use Spectral Decomposition and Write What the Expressions Are for the Operators

Multi-degree-of-freedom systems: Free vibrations

John T. Katsikadelis , in Dynamic Analysis of Structures, 2020

12.5.3 The generalized eigenvalue problem

The problem of determining the eigenvalues and eigenvectors of a matrix as stated above represents the typical or standard eigenvalue problem. However, the eigenvalue problem for determining the eigenfrequencies and mode shapes has a more general form

(12.5.54) $\left(\mathbf{A}-\lambda \mathbf{B}\right)\mathbf{x}=\mathbf{0}$

This problem is known as the generalized eigenvalue problem of linear algebra. In the literature, it is also referred to as the linearized eigenvalue problem. The study of the properties of the eigenvalues and eigenvectors of the generalized eigenvalue problem, Eq. (12.5.54), is facilitated if it is transformed to the standard eigenvalue problem, Eq. (12.5.6). Thus, the properties that apply to the standard eigenvalue problem can be transferred to the generalized eigenvalue problem. Without excluding the generality, the discussion will be restricted to real, symmetric, and positive definite matrices $\mathbf{A}$ and $\mathbf{B}$ because in free vibrations, they represent the stiffness and mass matrices, that is, $\mathbf{A}=\mathbf{K}$ and $\mathbf{B}=\mathbf{M}$ . Their positive definiteness results from the fact that the elastic energy $U\left(\mathbf{u}\right)$ and the kinetic energy $T\left(\stackrel{.}{\mathbf{u}}\right)$ are expressed by positive definite quadratic forms, that is,

(12.5.55) $U\left(\mathbf{u}\right)=\left\{\begin{array}{c}\frac{1}{2}{\mathbf{u}}^T\mathbf{Ku}>0,\text{if}\mathbf{u}\ne 0\\ \frac{1}{2}{\mathbf{u}}^T\mathbf{Ku}=0,\text{if}\mathbf{u}=0\end{array}\right.$

(12.5.56) $T\left(\stackrel{.}{\mathbf{u}}\right)=\left\{\begin{array}{c}\frac{1}{2}{\stackrel{.}{\mathbf{u}}}^T\mathbf{M}\stackrel{.}{\mathbf{u}}>0,\text{if}\stackrel{.}{\mathbf{u}}\ne 0\\ \frac{1}{2}{\stackrel{.}{\mathbf{u}}}^T\mathbf{M}\stackrel{.}{\mathbf{u}}=0,\text{if}\stackrel{.}{\mathbf{u}}=0\end{array}\right.$

Applying Eq. (12.5.49) to matrix $\mathbf{B}$ we have

(12.5.57) $\mathbf{B}=\tilde{\mathbf{X}}\tilde{\mathbf{\Lambda }}{\tilde{\mathbf{X}}}^T$

where $\tilde{\mathbf{\Lambda }}$ is the diagonal matrix of the eigenvalues of $\mathbf{B}$ and $\tilde{\mathbf{X}}$ the matrix of its eigenvectors. Obviously, when the matrix $\mathbf{B}$ is diagonal, as in the case of concentrated masses, then $\tilde{\mathbf{X}}=\mathbf{I}$ .

Further, we can set

(12.5.58) $\mathbf{B}={\mathbf{SS}}^T$

where

(12.5.59) $\mathbf{S}=\tilde{\mathbf{X}}\sqrt{\tilde{\mathbf{\Lambda }}}$

The matrix $\mathbf{S}$ is real on account that the elements of $\tilde{\mathbf{\Lambda }}$ are positive because $\mathbf{B}$ was assumed positive definite.

Substituting Eq. (12.5.59) into Eq. (12.5.54), yields

(12.5.60) $\mathbf{Ax}=\lambda {\mathbf{SS}}^T\mathbf{x}$

We define now the vector

(12.5.61) $\hat{\mathbf{x}}={\mathbf{S}}^T\mathbf{x}$

then ^b

(12.5.62) $\mathbf{x}={\mathbf{S}}^{-T}\hat{\mathbf{x}}$

Premultiplying Eq. (12.5.60) by ${\mathbf{S}}^{-1}$ and using Eq. (12.5.62) yield

(12.5.63) $\left(\hat{\mathbf{A}}-\lambda \mathbf{I}\right)\hat{\mathbf{x}}=0$

where

(12.5.64) $\hat{\mathbf{A}}={\mathbf{S}}^{-1}{\mathbf{AS}}^{-T}$

Obviously, the matrix $\hat{\mathbf{A}}$ is real and symmetric. Therefore, according to property 3, its eigenvalues and eigenvectors $\hat{\mathbf{x}}$ are real. Moreover, taking into account that the vectors $\hat{\mathbf{x}}$ satisfy the orthogonality condition, ${\hat{\mathbf{x}}}_i^T{\hat{\mathbf{x}}}_j=0$ , $i\ne j$ , we obtain.

(12.5.65) $\begin{array}{c}{\hat{\mathbf{x}}}_i^T{\hat{\mathbf{x}}}_j={\left({\mathbf{S}}^T{\mathbf{x}}_i\right)}^T{\mathbf{S}}^T{\mathbf{x}}_j\\ ={\mathbf{x}}_i^T{\mathbf{SS}}^T{\mathbf{x}}_j,i\ne j\\ ={\mathbf{x}}_i^T{\mathbf{Bx}}_j\\ =0\end{array}$

which implies that the eigenvectors of the generalized eigenvalue problem are orthogonal with respect to the matrix $\mathbf{B}$ .

The eigenvalue problems (12.5.54) and (12.5.63) have the same eigenvalues. Indeed, we can set $\mathbf{I}={\mathbf{S}}^{-1}{\mathbf{SS}}^T{\mathbf{S}}^{-T}$ and write Eq. (12.5.63) by virtue of Eqs. (12.5.64) and (12.5.58) as

(12.5.66) $\left[{\mathbf{S}}^{-1}\left(\mathbf{A}-\lambda \mathbf{B}\right){\mathbf{S}}^{-T}\right]\hat{\mathbf{x}}=\mathbf{0}$

If $\mathrm{\Pi }\left(\lambda \right)$ and $\hat{\mathrm{\Pi }}\left(\lambda \right)$ are the characteristic polynomials of the eigenvalue problems (12.5.54) and (12.5.63), respectively, we obtain

(12.5.67) $\begin{array}{c}\hat{\mathrm{\Pi }}\left(\lambda \right)=det\left(\hat{\mathbf{A}}-\lambda \mathbf{I}\right)\\ =det\left[{\mathbf{S}}^{-1}\left(\mathbf{A}-\lambda \mathbf{B}\right){\mathbf{S}}^{-T}\right]\\ =det\left({\mathbf{S}}^{-1}\right)det\left(\mathbf{A}-\lambda \mathbf{B}\right)det\left({\mathbf{S}}^{-T}\right)\\ =det\left({\mathbf{S}}^{-1}\right)\mathrm{\Pi }\left(\lambda \right)det\left({\mathbf{S}}^{-T}\right)\end{array}$

Because $det\left({\mathbf{S}}^{-1}\right)\ne 0$ and $det\left({\mathbf{S}}^{-T}\right)\ne 0$ , it implies that both eigenvalue problems have the same characteristic equation, hence the same eigenvalues.

The spectral decomposition of matrix $\mathbf{B}$ requires the complete solution of the eigenvalue problem. Therefore, the transformation of the generalized eigenvalue problem on the basis of Eq. (12.5.59) is not the most convenient one. A usual method to determine the matrix $\mathbf{S}$ is the Cholesky decomposition method, or the square root method, in which the matrix $\mathbf{B}$ is written in the form of a product, that is,

(12.5.68) $\mathbf{B}={\mathbf{U}}^T\mathbf{U}$

where $\mathbf{U}$ is an upper triangular matrix. Hence

(12.5.69) $\mathbf{S}={\mathbf{U}}^T$

9.

If the matrices $\mathbf{A}$ and $\mathbf{B}$ real, symmetric, and positive definite, then the generalized eigenvalue problem has positive eigenvalues.

This is readily proved by premultiplying Eq. (12.5.54) by ${\mathbf{x}}^T$ . This gives

(12.5.70) ${\mathbf{x}}^T\mathbf{Ax}=\lambda {\mathbf{x}}^T\mathbf{Bx}$

Because $\mathbf{A}$ and $\mathbf{B}$ are positive definite, we obtain

(12.5.71) $\lambda =\frac{{\mathbf{x}}^T\mathbf{Ax}}{{\mathbf{x}}^T\mathbf{Bx}}>0$

10.

The eigenvectors ${\mathbf{x}}_i$ of the generalized eigenvalue problem are orthogonal with respect to the matrices $\mathbf{A}$ and $\mathbf{B}$ .

It was previously shown that

(12.5.72) ${\mathbf{x}}_i^T{\mathbf{Bx}}_j=0,i\ne j$

Consequently, we obtain

(12.5.73) $\begin{array}{c}{\mathbf{x}}_i^T{\mathbf{Ax}}_j=\lambda {\mathbf{x}}_i^T{\mathbf{Bx}}_j\\ =0\end{array}$

Example 12.5.3

Transform the generalized eigenvalue problem into a standard eigenvalue problem using the spectral decomposition method, when

$\mathbf{A}=\left[\begin{array}{cc}50& -40\\ -40& 90\end{array}\right],\mathbf{B}=\left[\begin{array}{ll}2.5& -1\hfill \\ -1& 3.2\hfill \end{array}\right]$

Solution

The characteristic equation of $\mathbf{B}$ is

(1) $\left|\begin{array}{ll}2.5-\tilde{\lambda }& -1\hfill \\ -1& 3.2-\tilde{\lambda }\end{array}\right|={\tilde{\lambda }}^2-5.7\tilde{\lambda }+7.0=0$

from which we obtain

(2) ${\tilde{\lambda }}_1=1.790,{\tilde{\lambda }}_2=3.909$

Hence

(3) $\tilde{\mathbf{\Lambda }}\mathbf{=}\left[\begin{array}{ll}1.790& 0\hfill \\ 0& 3.909\end{array}\right]$

The eigenvectors are computed from the solution of the homogeneous linear system

(4) $\left[\begin{array}{ll}2.5-\tilde{\lambda }& -1\hfill \\ -1& 3.2-\tilde{\lambda }\hfill \end{array}\right]\left\{\begin{array}{c}{x}_1\\ x_2\end{array}\right\}=\left\{\begin{array}{c}0\\ 0\end{array}\right\}$

for ${\tilde{\lambda }}_1=1.790$ and ${\tilde{\lambda }}_2=3.909$ .

Thus, we obtain the matrix of the eigenvectors normalized with respect to their magnitude

(5) $\tilde{\mathbf{X}}=\left[\begin{array}{cc}0.8157& 0.5784\\ 0.5785& -0.8158\end{array}\right]$

Using Eq. (12.5.59) we obtain

(6) $\mathbf{S}=\tilde{\mathbf{X}}\sqrt{\tilde{\mathbf{\Lambda }}}=\left[\begin{array}{cc}1.0913& 1.1435\\ 0.7740& -1.6129\end{array}\right],{\mathbf{S}}^{-1}=\left[\begin{array}{cc}0.6097& 0.4323\\ 0.2926& -0.4126\end{array}\right]$

and on the basis of Eq. (12.5.64)

(7) $\hat{\mathbf{A}}={\mathbf{S}}^{-1}{\mathbf{AS}}^{-T}=\left[\begin{array}{ll}14.3213& -2.1288\hfill \\ -2.1288& 29.2564\hfill \end{array}\right]$

11.

If the real and symmetric matrix $\mathbf{A}$ is singular, then the generalized eigenvalue problem has at least one zero eigenvalue and the corresponding eigenvector is different from zero.

First, we will show that this property holds for the standard eigenvalue problem $\mathbf{Ax}=\lambda \mathbf{x}$ . For this purpose, we write $\mathbf{A}$ in the form of its spectral decomposition,

(12.5.74) $\mathbf{A}=\mathbf{X\Lambda }{\mathbf{X}}^T$

Because it was assumed that $\mathbf{A}$ is singular, it implies that $det\left(\mathbf{A}\right)=0$ . Moreover, Eq. (12.5.47) gives $det\left({\mathbf{X}}^T\right)det\left(\mathbf{X}\right)=1$ , hence

(12.5.75) $\begin{array}{c}det\left(\mathbf{A}\right)=det\left({\mathbf{X}}^T\right)det\left(\mathbf{\Lambda }\right)det\left(\mathbf{X}\right)\\ =det\left(\mathbf{\Lambda }\right)\\ ={\lambda }_1{\lambda }_2\cdots {\lambda }_N\\ =0\end{array}$

from which we conclude that at least one of the ${\lambda }_i$ is zero and the eigenvalue problem for this value becomes

(12.5.76) ${\mathbf{Ax}}_i=\mathbf{0}$

which yields ${\mathbf{x}}_i\ne \mathbf{0}$ because $det\left(\mathbf{A}\right)=0$ .

The generalized eigenvalue problem $\mathbf{Ax}=\lambda \mathbf{Bx}$ is transformed to the standard eigenvalue problem $\hat{\mathbf{A}}\hat{\mathbf{x}}=\lambda \hat{\mathbf{x}}$ , hence it is

(12.5.77) $det\left(\hat{\mathbf{A}}\right)={\lambda }_1{\lambda }_2\cdots {\lambda }_N$

which by virtue of Eq. (12.5.64) gives

(12.5.78) $\begin{array}{c}det\left(\hat{\mathbf{A}}\right)=det\left({\mathbf{S}}^{-1}\right)det\left(\mathbf{A}\right)det\left({\mathbf{S}}^{-T}\right)\\ =0\end{array}$

because it was assumed $det\left(\mathbf{A}\right)=0$ . Eqs. (12.5.77) and (12.5.78) imply that at least one of the eigenvalues ${\lambda }_i$ is zero. Moreover, it is ${\hat{\mathbf{x}}}_i\ne 0$ and by virtue of Eq. (12.5.62), we obtain ${\mathbf{x}}_i={\mathbf{S}}^{-T}{\hat{\mathbf{x}}}_i\ne \mathbf{0}$ .

If the matrices $\mathbf{A}$ and $\mathbf{B}$ represent the stiffness and mass matrices of the structure, that is, $\mathbf{A}=\mathbf{K}$ , $\mathbf{B}=\mathbf{M}$ , then the eigenvector ${\mathbf{x}}_i$ corresponding to the zero eigenvalue represents rigid body motion. This is shown right below.

If we set ${\mathbf{x}}_i={\mathbf{u}}_i$ , then ${\mathbf{Ku}}_i$ represents the vector of the elastic force ${\mathbf{f}}_{\mathit{Si}}$ corresponding to the displacement ${\mathbf{u}}_i$ , that is

(12.5.79) ${\mathbf{f}}_{\mathit{Si}}={\mathbf{Ku}}_i$

or because ${\lambda }_i=0$

(12.5.80) $\begin{array}{c}{\mathbf{Ku}}_i={\lambda }_i{\mathbf{Mu}}_i\\ =0\end{array}$

Hence ${\mathbf{f}}_{\mathit{Si}}=\mathbf{0}$ while Eq. (12.5.76) yields ${\mathbf{u}}_i\ne 0$ , which is due to the motion of the structure as a rigid body.

12.

If the real and symmetric matrix $\mathbf{B}$ is singular, then the generalized eigenvalue problem has at least one infinite eigenvalue.

This is shown if the eigenvalue problem is written in the form

(12.5.81) $\mathbf{Bx}=\mu \mathbf{Ax}$

where $\mu =1/\lambda$ .

13.

Any vector $\mathbf{u}$ with a dimension $N$ can be represented as the superposition of the eigenvectors of the eigenvalue problem.

We showed that the set of the eigenvectors ${\mathbf{x}}_1,{\mathbf{x}}_2,\dots ,{\mathbf{x}}_N$ of an $N\times N$ symmetric matrix is linearly independent and can be employed as a base of the $N$ dimensional space to represent an arbitrary vector $\mathbf{u}$ in that space. Thus we may set

(12.5.82) $\mathbf{u}=a_1{\mathbf{x}}_1+a_2{\mathbf{x}}_2+\cdots +a_N{\mathbf{x}}_N$

or

(12.5.83) $\mathbf{u}=\mathbf{Xa}$

where $\mathbf{X}$ is the matrix of the eigenvectors and $\mathbf{a}={\left\{a_1,a_2,\dots ,a_N\right\}}^T$ the vector of the coefficients. The matrix $\mathbf{X}$ is not singular because the eigenvectors ${\mathbf{x}}_i$ are linearly independent. Hence

(12.5.84) $\mathbf{a}={\mathbf{X}}^{-1}\mathbf{u}$

or using Eq. (12.5.48)

(12.5.85) $\mathbf{a}={\mathbf{X}}^T\mathbf{u}$

or

(12.5.86) $a_i={{\mathbf{x}}_i}^T\mathbf{u},i=1,2,\dots ,N$

Similarly, we can use the eigenvectors of the generalized eigenvalue problem to represent the vector $\mathbf{u}$ . The establishment of the coefficients $a_i$ in that case is established by premultiplying both sides of Eq. (12.5.82) by ${\mathbf{x}}_i^T\mathbf{B}$ and noting that ${\mathbf{x}}_i^T{\mathbf{Bx}}_j=0$ for $i\ne j$ . This yields

(12.5.87) $a_i=\frac{{\mathbf{x}}_i^T\mathbf{Bu}}{{\mathbf{x}}_i^T{\mathbf{Bx}}_i}$

If the eigenvectors ${\mathbf{x}}_i$ are normalized with respect to $\mathbf{B}$ so that ${\mathbf{x}}_i^T{\mathbf{Bx}}_i=1$ , then Eq. (12.5.87) gives

(12.5.88) $a_i={\mathbf{x}}_i^T\mathbf{Bu},i=1,2,\dots ,N$

or

(12.5.89) $\mathbf{a}={\mathbf{X}}^T\mathbf{Bu}$

The representation of a vector as a superposition of the eigenvectors as in Eq. (12.5.82) is known as the expansion theorem. As we will see in Section 14.5, this theorem is a special case of a Ritz vector representation when the eigenvectors are used as Ritz vectors.

14.

If a real and symmetric matrix $\mathbf{A}$ is singular, then the quadratic form $\mathrm{\Pi }\left(\mathbf{u}\right)={\mathbf{u}}^T\mathbf{Au}$ is positive semidefinite, that is, $\mathrm{\Pi }\left(\mathbf{u}\right)={\mathbf{u}}^T\mathbf{Au}\ge 0$ for $\mathbf{u}\ne 0$ .

We write the vector $\mathbf{u}$ in the form of Eq. (12.5.83) and the matrix $\mathbf{A}$ in the form of Eq. (12.5.49). Thus we have

(12.5.90) $\begin{array}{c}\mathrm{\Pi }\left(\mathbf{u}\right)={\mathbf{u}}^T\mathbf{Au}\\ ={\mathbf{a}}^T{\mathbf{X}}^T\mathbf{AXa}\\ ={\mathbf{a}}^T\mathbf{\Lambda a}\\ =\sum _{i=1}^N{\lambda }_i{a}_i^2\end{array}$

Because $\mathbf{A}$ is singular, at least on of its eigenvalues is zero, say ${\lambda }_k=0$ . If we take $\mathbf{u}={\mathbf{x}}_k$ , it will be $a_k\ne 0$ , $a_{i\ne k}=0$ and Eq. (12.5.90) becomes

(12.5.91) $\begin{array}{c}\mathrm{\Pi }\left(\mathbf{u}\right)={\lambda }_k{a}_k^2\\ =0\end{array}$

Therefore

(12.5.92) $\mathrm{\Pi }\left(\mathbf{u}\right)={\mathbf{u}}^T\mathbf{Au}\ge 0\text{for}\mathbf{u}\ne 0$

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Phase-field modeling of fracture

Jian-Ying Wu , ... Stéphane P.A. Bordas , in Advances in Applied Mechanics, 2020

B.1 General definition of the positive/negative projection

Let us consider the following spectral decomposition of the specified second-order tensor A

(B.1) $\begin{array}{l}\hfill \mathit{A}=\sum _{n=1}^3{A}_n{\mathit{p}}_n\otimes {\mathit{p}}_n={\mathit{A}}^{+}+{\mathit{A}}^{-},\end{array}$

for the eigenvalues A _n = P _nn : A = p _n ⋅ A ⋅ p _n and the associated eigenvectors p _n , respectively, with the latter defining the second-order symmetric tensor P _nn := p _n ⊗ p _n . Note that the subscript n does not refer to the dummy index of a tensor (or vector), but rather, it represents the quantity associated with the n-th principal value.

In Eq. (B.1) the positive/negative components A ^± are coaxial to the parent tensor A . Without loss of generality, they are expressed as

(B.2) $\begin{array}{l}\hfill {\mathit{A}}^{+}=\sum _n{A}_n^{+}{\mathit{P}}_{nn}={\mathbb{P}}^{+}:\mathit{A},{\mathit{A}}^{-}=\mathit{A}-{\mathit{A}}^{+}=\sum _n{A}_n^{-}{\mathit{P}}_{nn}={\mathbb{P}}^{-}:\mathit{A}\end{array}$

where $A_n^{+}={\mathit{P}}_{nn}^{+}:\mathit{A}\ge 0$ and $A_n^{-}:=A_n-A_n^{+}={\mathit{P}}_{nn}^{-}:\mathit{A}\le 0$ are the principal values of the positive component A ⁺ and the negative one A ⁻, respectively; the fourth-order projection operators ${\mathbb{P}}^{\pm }$ are given by

(B.3) $\begin{array}{l}\hfill {\mathbb{P}}^{+}=\sum _n{\mathit{P}}_{nn}\otimes {\mathit{P}}_{nn}^{\pm },{\mathbb{P}}^{-}=\mathbb{I}-{\mathbb{P}}^{+}\end{array}$

with the second-order symmetric tensors ${\mathit{P}}_{nn}^{\pm }$ dependent on the adopted PNP scheme. Note that the projection operators satisfy the idempotent property, i.e.,

(B.4) $\begin{array}{l}\hfill {\mathbb{P}}^{\pm }:{\mathbb{P}}^{\pm }:\mathit{A}={\mathbb{P}}^{\pm }:{\mathit{A}}^{\pm }={\mathit{A}}^{\pm }⟹{\mathbb{P}}^{+}:{\mathit{A}}^{-}={\mathbb{P}}^{-}:{\mathit{A}}^{+}=\mathit{0}\hfill \end{array}$

That is, the positive/negative decomposition is an orthogonal projection. Furthermore, it also follows that

(B.5) $\begin{array}{l}\hfill \frac{\text{d}{\mathit{A}}^{\pm }}{\text{d}\mathit{A}}={\mathbb{Q}}^{\pm }\end{array}$

where the fourth-order tensor ${\mathbb{Q}}^{\pm }$ are expressed as (Wu & Xu, 2013)

(B.6) $\begin{array}{l}\hfill {\mathbb{Q}}^{+}={\mathbb{P}}^{+}+2\sum _n\sum _{m>n}\frac{A_n^{+}-A_m^{+}}{A_n-A_m}{\mathit{P}}_{nm}\otimes {\mathit{P}}_{nm},{\mathbb{Q}}^{-}=\mathbb{I}-{\mathbb{Q}}^{+}\end{array}$

for the second-order symmetric tensor ${\mathit{P}}_{nm}={({\mathit{p}}_n\otimes {\mathit{p}}_m)}^{\text{s}}$ . Owing to the orthogonal property P _nm : A = 0 for m ≠ n, the following identities hold

(B.7) $\begin{array}{l}\hfill {\mathbb{Q}}^{\pm }:\mathit{A}={\mathbb{P}}^{\pm }:\mathit{A}={\mathit{A}}^{\pm },{\dot{\mathbb{Q}}}^{\pm }:\mathit{A}={\dot{\mathit{A}}}^{\pm }-{\mathbb{Q}}^{\pm }:\dot{\mathit{A}}=\mathit{0}\end{array}$

That is, the fourth-order tensors ${\mathbb{Q}}^{\pm }$ extract the same positive/negative components A ^± as the irreducible PNP operators ${\mathbb{P}}^{\pm }$ do. In the following, the classical positive/negative projection and the novel one are briefly introduced.

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Method of Moments and Fast Algorithms

Christophe Bourlier , in Radar Propagation Modeling in a Complex Maritime Environment, 2018

1.4.2.2 Decomposition of H₀ ⁽¹⁾ as a superposition of plane waves

From the Weyl decomposition (spectral decomposition), the Hankel function can be written as follows:

[1.42] $\begin{array}{c}{\mathrm{H}}_0^{\left(1\right)}\left(k_0‖{\mathit{r}}_m-{\mathit{r}}_n‖\right)=\frac{1}{\pi }{\int }_{C_g}{e}^{{\mathit{jk}}_0\left[\left(x_m-x_n\right)\text{cos}\varphi +\left(z_m-z_n\right)\text{sin}\varphi \right]}d\varphi \\ =\frac{1}{\pi }{\int }_{C_g}{e}^{{\mathit{jk}}_0‖{\mathbf{r}}_m-{\mathbf{r}}_n‖\text{cos}\left(\varphi -{\varphi }_s\right)}d\varphi ,\end{array}$

where the integration path $C_g=]-\frac{\pi }{2}+j\infty ;-\frac{\pi }{2}\left[\cup \left[-\frac{\pi }{2};+\frac{\pi }{2}\right]\cup \right]+\frac{\pi }{2};+\frac{\pi }{2}-j\infty [$ is shown in Figure 1.2 in black full line.

Figure 1.2. Top: integration path C _g of the Hankel function and the modified one, C _δ . Bottom: physical interpretation of C _δ in the spatial domain. For a color version of this figure, see www.iste.co.uk/bourlier/radar.zip

As shown at the top (spectral domain) of Figure 1.2, the SA method substitutes the integration contour C _g for a unique steepest descent path C _δ going through the origin. This path results from a group of paths {C _m,n } associated with all couples of points ( r _m , r _n ) going through the saddle points $\left\{{\varphi }_s^{m,n}=\text{arctan}\left(\frac{z_m-z_n}{x_m-x_n}\right)\right\}\in \left[-\frac{\pi }{2};+\frac{\pi }{2}\right]$ (blue bold points). Furthermore, close to the origin, C _δ is a straight line having a slope − tan δ. If δ is correctly chosen, the integrand of [1.42] decays rapidly away from the origin and the phase has little variations. Thus, as in a classical saddle-point technique, after changing the original path C _g by C _δ in equation [1.42], the integration over ϕ = ϕ _R (1 – j tan δ) can be approximated by a sum over a limited number of complex angles, where ${\varphi }_R\in \left[-{\varphi }_{\text{max}};+{\varphi }_{\text{max}}\right]\in \mathrm{ℝ}$ ; the angle ϕ _max being an upper limit of ϕ _s,max.

In conclusion, sampling integral [1.42] with respect to the path C _δ given by ${\varphi }_R\left(1-j\text{tan}\delta \right)={\varphi }_R{e}^{-j\delta }/\text{cos}\left(\delta \right)\left({\varphi }_R\in \mathrm{ℝ}\right)$ , the Hankel function can be evaluated as follows:

[1.43] $\begin{array}{l}{\mathrm{H}}_0^{\left(1\right)}\left(k_0‖{\mathit{r}}_m-{\mathit{r}}_n‖\right)\approx \frac{e^{-j\delta }}{\pi \text{cos}\left(\delta \right)}\\ \times \sum _{q=-Q}^{q=+Q}{e}^{{\mathit{jk}}_0\left[\left(x_m-x_n\right)\text{cos}\varphi +\left(z_m-z_n\right)\text{sin}\varphi \right]}\Delta \varphi ,\end{array}$

where $\Delta {\varphi }_R=2{\varphi }_{\text{max}}/\left(2Q+1\right)\in \mathrm{ℝ}$ and ϕ  = qΔϕ _R (1   − j  tan δ). Equation [1.43] means that the Hankel function can be approximated by 2Q + 1 plane waves of complex propagation angles.

As shown at the bottom of Figure 1.2, ϕ _s,max corresponds to the maximum angle defined with respect to x , for which the current point sees the other points on the surface. For this region, the angle ϕ _m,n is close to the saddle point ϕ _s ^m,n , and the imaginary part of ϕ _m,n is small. The associated waves are propagated. On the other hand, if ${\varphi }_{m,n}>{\varphi }_{s,\text{max}}\in \mathrm{ℝ}$ , the imaginary part of ϕ _m,n becomes larger and the associated waves are not propagated (evanescent waves). This corresponds to the shadowed zone or the weak interaction zone.

The substitution of equation [1.43] into equation [1.40] with Z _{m, n}   =   H₀ ⁽¹⁾(k ₀‖ r _m   − r _n ‖) leads for the forward weak interactions to

[1.44] $Y_m^{\text{Weak}}\approx \frac{e^{-j\delta }}{\pi \text{cos}\delta }\sum _{q=-Q}^{q=+Q}{W}_m\left(\varphi \right)e^{{\mathit{jk}}_0{z}_m\text{sin}{\varphi }_p}\Delta {\varphi }_R,$

where

[1.45] $\begin{array}{l}{W}_m\left(\varphi \right)=\sum _{n=1}^{m-N_{\text{Strong}}-1}{X}_n{e}^{{\mathit{jk}}_0\left[\left(x_m-x_n\right)\text{cos}\varphi -z_n\text{sin}\varphi \right]}\\ =\sum _{n=1}^{m-N_{\text{Strong}}-2}{X}_n{e}^{{\mathit{jk}}_0\left[\left(x_m-x_n\right)\text{cos}\varphi -z_n\text{sin}\varphi \right]}\\ +X_{m_0}{e}^{{\mathit{jk}}_0\left[\left(x_m-x_{m_0}\right)\text{cos}\varphi -z_{m_0}\text{sin}\varphi \right]}\\ =W_{m-1}\left(\varphi \right)e^{{\mathit{jk}}_0\Delta x\text{cos}\varphi }\\ +X_{m_0}{e}^{{\mathit{jk}}_0\left(N_{\text{Strong}}+1\right)\text{cos}\varphi \Delta x}{e}^{-{\mathit{jk}}_0{z}_{m_0}\text{sin}\varphi },\end{array}$

where x _m = x _{m − 1} + Δx and x _m   − x _{m 0}   =   (N _Strong  +   1)Δx, in which m ₀ = m − N _Strong − 1.

In conclusion, a row of the matrix–vector product is expressed as the sum over Q since there exists a recurrence relation between W _m and W _{m  −   1}. From equation [1.38], Y _m must be multiplied by $j\left|{\Delta }_m\right|\sqrt{1+{\gamma }_m^2}/4$ .

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Thermodynamic model of SMA pseudoelasticity based on multiplicative decomposition of deformation gradient tensor

Andrzej Ziółkowski , in Pseudoelasticity of Shape Memory Alloys, 2015

7.3 Strain measures at finite deformations

The following relations result from polar and spectral decomposition of tensor of elastic deformation gradient

(7.5) ${\mathit{F}}^{\mathrm{e}}={\mathit{R}}^{\mathrm{e}}{\mathit{U}}^{\mathrm{e}}={\mathit{V}}^{\mathrm{e}}{\mathit{R}}^{\mathrm{e}},{\mathit{U}}^{\mathrm{e}}={\displaystyle \sum {\lambda }_i^{\mathrm{e}}}{\mathsf{N}}_i^{\mathrm{e}}\otimes {\mathsf{N}}_i^{\mathrm{e}},{\mathit{V}}^{\mathrm{e}}={\displaystyle \sum {\lambda }_i^{\mathrm{e}}}{\mathsf{n}}_i^{\mathrm{e}}\otimes {\mathsf{n}}_i^{\mathrm{e}}\text{,}$

where R ^e is proper orthogonal rotation tensor. The positive definite, symmetric tensors U ^e, V ^e are right and left elastic stretch tensors, respectively. The scalars λ _i ^e are the principal elastic stretches common for tensors U ^e and V ^e. They are eigenvalues of the characteristic equation $det\left({\mathit{F}}^{\mathrm{e}}{}^T\mathit{F}^{\mathrm{e}}-{\lambda }^2\mathbf{I}\right)=0$ . Corresponding to them, eigenvectors ${\mathsf{N}}_i^{\mathrm{e}}$ and ${\mathsf{n}}_i^{\mathrm{e}}$ are Lagrangean (material) and Eulerian (spatial) principal directions of elastic stretch tensors U ^e and V ^e. Principal directions vectors ${\mathsf{N}}_i^{\mathrm{e}}$ can be rotated to principal directions ${\mathsf{n}}_i^{\mathrm{e}}$ of tensor V ^e with the aid of rotation tensor R ^e _.

(7.6) ${\mathsf{n}}_i^{\mathrm{e}}={\mathit{R}}^{\mathrm{e}}{\mathsf{N}}_i^{\mathrm{e}},{\mathit{R}}^{\mathrm{e}}={\mathsf{n}}_i^{\mathrm{e}}\otimes {\mathsf{N}}_i^{\mathrm{e}},{\left({\mathit{R}}^{\mathrm{e}}\right)}^{\mathrm{T}}{\mathit{R}}^{\mathrm{e}}=\mathbf{I},det\left({\mathit{R}}^{\mathrm{e}}\right)=\mathbf{I}\text{.}$

The logarithmic (Hencky) strain measure is used in the present model as a measure of elastic deformation. In mobile Lagrangean description, this measure is based on tensor U ^e, while in Eulerian description on tensor V ^e

(7.7) $\begin{array}{l}{\mathit{E}}^{\mathrm{e}}\left(0\right)=ln\left({\mathit{U}}^{\mathrm{e}}\right)={\displaystyle \sum ln\left({\lambda }_i^{\mathrm{e}}\right)}{\mathsf{N}}_i^{\mathrm{e}}\otimes {\mathsf{N}}_i^{\mathrm{e}},{\mathit{e}}^{\mathrm{e}}\left(0\right)=ln\left({\mathit{V}}^{\mathrm{e}}\right)={\displaystyle \sum ln\left({\lambda }_i^{\mathrm{e}}\right)}{\mathsf{n}}_i^{\mathrm{e}}\otimes {\mathsf{n}}_i^{\mathrm{e}},\hfill \\ {\mathit{e}}^{\mathrm{e}}\left(0\right)={\mathit{R}}^{\mathrm{e}}{\mathit{E}}^{\mathrm{e}}\left(0\right){\left({\mathit{R}}^{\mathrm{e}}\right)}^{\mathrm{T}}\hfill \end{array}$

The Lagrangean (material) elastic strain measure E ^e(0) is connected with its counterpart Eulerian (spatial) elastic strain e ^e(0) by formula (7.7) ₃. Strain measure E ^e(0) is an element of parametric family of Lagrangean elastic strain measures—for parameter $n=0$ , analogical to the family of total strain measures introduced originally by Hill (1978),

(7.8) $\begin{array}{l}{\mathit{E}}^{\mathrm{e}}\left(n\right)=\left({\left({\mathit{U}}^{\mathrm{e}}\right)}^{2n}-\mathbf{I}\right)/\left(2n\right);n\ne 0,{\mathit{E}}^{\mathrm{e}}\left(0\right)=ln\left({\mathit{U}}^{\mathrm{e}}\right);n=0,\hfill \\ {\mathit{E}}^{\mathrm{e}}\left(n\right)={\displaystyle \sum f\left({\lambda }_i^{\mathrm{e}}\right)}{\mathsf{N}}_i^{\mathrm{e}}\otimes {\mathsf{N}}_i^{\mathrm{e}},\hfill \\ f\left({\lambda }_i^{\mathrm{e}}\right)=\left({\left({\lambda }_i^{\mathrm{e}}\right)}^{2n}-1\right)/2n,n\ne 0,f\left({\lambda }_i^{\mathrm{e}}\right)=ln\left({\lambda }_i^{\mathrm{e}}\right),n=0\text{.}\hfill \end{array}$

The scale function f(λ _i ^e) of principal elastic stretches has the properties $f\left({\lambda }_i^{\mathrm{e}}=1\right)=0$ and $\mathrm{d}f\left({\lambda }_i^{\mathrm{e}}=1\right)/\mathrm{d}{\lambda }_i^{\mathrm{e}}=1$ . These properties assure that all strain measures from the family (7.8) converge to small strains tensor ${\mathit{\varepsilon }}^{\mathrm{e}}\equiv 0.5\left({\mathit{F}}^{\mathrm{e}}+{}^T\mathit{F}^{\mathrm{e}}-2\mathbf{I}\right)$ when elastic deformation gradient tends to unity $\left({\mathit{F}}^{\mathrm{e}}-\mathbf{I}\right)\to \mathbf{0}$ .

The logarithmic elastic strain measure E ^e(0) possesses the known, valuable property that its spherical part describes purely dilatational deformation of the material element, while its deviatoric part describes purely distortional deformation at finite deformations

(7.9) $\begin{array}{l}{\mathit{E}}^{\mathrm{e}}\left(0\right)=ln\left({\mathit{U}}^{\mathrm{e}}\right)={\tilde{\mathit{E}}}^{\mathrm{e}}\left(0\right)+\frac{1}{3}\mathrm{t}\mathrm{r}\left({\mathit{E}}^{\mathrm{e}}\left(0\right)\right)\mathbf{I},E_{ij}^{\mathrm{e}}=ln\left({\lambda }_i^{\mathrm{e}}\right)={\tilde{E}}_{ij}^{\mathrm{e}}+\frac{1}{3}ln\left({\lambda }_1^{\mathrm{e}}{\lambda }_2^{\mathrm{e}}{\lambda }_3^{\mathrm{e}}\right){\delta }_{ij},\hfill \\ \mathrm{t}\mathrm{r}\left({\mathit{E}}^{\mathrm{e}}\left(0\right)\right)=ln\left({\lambda }_1^{\mathrm{e}}\right)+ln\left({\lambda }_2^{\mathrm{e}}\right)+ln\left({\lambda }_3^{\mathrm{e}}\right)=ln\left(J^{\mathrm{e}}\right),\hfill \\ J^{\mathrm{e}}\equiv det\left({\mathit{F}}^{\mathrm{e}}\right)={\rho }^{\mathrm{in}}/\rho =det\left({\mathit{U}}^{\mathrm{e}}\right)={\lambda }_1^{\mathrm{e}}{\lambda }_2^{\mathrm{e}}{\lambda }_3^{\mathrm{e}}\text{,}\hfill \end{array}$

where ρ ⁱⁿ and ρ denote density in stress free ("natural") and actual configuration, respectively. The Ẽ ^e(0) is the deviator of logarithmic elastic strain $\left({\tilde{E}}_1^{\mathrm{e}}+{\tilde{E}}_2^{\mathrm{e}}+{\tilde{E}}_3^{\mathrm{e}}=0\right)$ .

It is this property, which is not possessed by other strain measures from the family (7.8), for example, the Green-Lagrange strain E ^e(1), that prevailed in selection of E ^e(0) as a state parameter in the SMA materials model discussed here. The principal values of elastic stretch tensor U ^e can be determined if there are known components of elastic logarithmic strain with the aid of the formula ${\lambda }_i^{\mathrm{e}}={\left(J^{\mathrm{e}}\right)}^{1/3}exp\left({\tilde{E}}_i^{\mathrm{e}}\right)$ .

The experiments show that under arbitrary loadings metallic materials exhibit only small distortional elastic strains, as they start to flow inelastically at strains of the order of 0.002. Only dilatational elastic strains can possibly be large, for example, at dynamic loadings due to high pressures. Hence, the special property of logarithmic strain measure E ^e(0) of uncoupling dilatational and distortional deformation effects makes this measure particularly useful in building constitutive models of metallic materials. It straightforwardly allows postulating independently constitutive relations for distortional and dilatational deformations of a material.

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On asymptotic structure of three-body scattering states for the scattering problem of charged quantum particles

A.M. Budylin , ... S.B. Levin , in Advances in Quantum Chemistry, 2021

4.2 An "uncertainty" fixation and a generating integral

In order to fix the uncertainty in the presentation of the spectral decomposition Ψ ^sep (X, P) (30), we will need a certain construction to be called a generating integral.

The essence of the construction is in our presenting a certain function ${\phi }_n(\mathbf{x},\hat{\mathit{\omega }})$ , for which there is a function $R_n(\mathbf{P},{\mathbf{p}}^{\prime },\hat{\mathit{\omega }})$ , the one with which the following equation is true

(31) $\begin{array}{l}\hfill {\int }_{{\mathbf{S}}^2}d\hat{\mathit{\omega }}{\phi }_n(\mathbf{x},\hat{\mathit{\omega }})R_n(\mathbf{P},\mathbf{p}\prime ,\hat{\mathit{\omega }})=\sum _{l=0}^{n-1}\sum _{m=-l}^l{\psi }_{nlm}(x)Y_l^m(\hat{\mathbf{x}})R_{nlm}(\mathbf{P},\mathbf{p}\prime ).\end{array}$

The function ${\phi }_n(\mathbf{x},\hat{\mathit{\omega }})$ is described in detail in Appendix A. Substitute the function defined in Eq. (A.14) into Eq. (31). It is easy to see that if we define the kernel $R_n(\mathbf{P},{\mathbf{p}}^{\prime },\hat{\mathit{\omega }})$ , in its terms we will define the kernels R _nlm (P, p′) having the meaning of decomposition coefficients of the Schrodinger operator continuous spectrum three-body eigenfunction in discrete spectrum eigenfunctions of a pairwise subsystem in the presentation of the spectral decomposition (30) type. To be exact, if we define a radial part of two-body Coulomb operator ψ _nlm (x) eigenfunctions in the standard way ⁵

$\begin{array}{l}\hfill {\psi }_{nlm}(x)=N_{nlm}{e}^{-\frac{|\alpha |}{2n}x}{x}^l\mathrm{\Phi }\left(-n+l+1,2l+2,\frac{|\alpha |}{n}x\right),\end{array}$

where N _nlm is a normalization constant depending only on quantum numbers, then according to (31) and (A.13), (A.14) we obtain the following expression for the kernels R _nlm :

(32) $\begin{array}{l}\hfill R_{nlm}(\mathbf{P},\mathbf{p}\prime )=\frac{1}{N_{nlm}}{\beta }_{nl}\frac{1}{(2l+1)!}{(-\frac{|c|}{n})}^l{\int }_{{\mathbf{S}}^2}d\hat{\mathit{\omega }}{R}_n(\mathbf{P},\mathbf{p}\prime ,\hat{\mathit{\omega }})Y_l^m(\hat{\mathit{\omega }}).\end{array}$

Having thus defined the "generating" function ${\phi }_n(\mathbf{x},\hat{\mathit{\omega }})$ , we replace a triple infinite sum in three quantum numbers by the single sum and the unit sphere integral in the spectral decomposition. Such a simplification turns quite significant, as it allows to change a search of a set of n ² unknown coefficients R _nlm (P, p′) by each fixed principal quantum number n for a search of one unknown function $R_n(\mathbf{P},{\mathbf{p}}^{\prime },\hat{\mathit{\omega }})$ . Hereafter we will be using a connection of the Laguerre polynomial and the confluent hypergeometric function with a whole negative first argument formed as follows:

(33) $\begin{array}{l}\hfill {\phi }_n(\mathbf{x},\hat{\mathbf{k}})=e^{-\frac{|c_1|}{2n}x}{L}_{n-1}\left(\frac{|c_1|}{n}x{sin}^2\frac{\theta }{2}\right),{sin}^2\theta /2=\frac{1-〈\hat{\mathbf{k}},\hat{\mathbf{x}}〉}{2}.\end{array}$

Here L _n (y)—are the Laguerre polynomials.

Thus a presentation for the function Ψ ^sep (X, P) (30) in new terms takes the form

(34) $\begin{array}{l}\hfill {\mathrm{\Psi }}^{sep}(\mathbf{X},\mathbf{P})={\int }_{{\mathbf{R}}^3}d{\mathbf{k}}^{\prime }{\int }_{{\mathbf{R}}^3}d{\mathbf{p}}^{\prime }{\psi }_c(\mathbf{x},{\mathbf{k}}^{\prime }){\psi }_c^{eff}(\mathbf{y},{\mathbf{p}}^{\prime })\delta ({k^{\prime }}^2+{p^{\prime }}^2-E)R(\mathbf{P},{\mathbf{P}}^{\prime })\\ & +\sum _{n=1}^{\infty }{\int }_{{\mathbf{S}}^2}d{\hat{\mathbf{k}}}^{\prime }{\int }_{{\mathbf{R}}^3}d{\mathbf{p}}^{\prime }{\phi }_n(\mathbf{x},{\hat{\mathbf{k}}}^{\prime }){\psi }_c^{eff}(\mathbf{y},{\mathbf{p}}^{\prime })\delta ({p^{\prime }}^2-\frac{c_1^2}{4n^2}-E)R_n(\mathbf{P},{\mathbf{p}}^{\prime },{\hat{\mathbf{k}}}^{\prime }).\end{array}$

Now we are ready to pass on to the procedure of reconstruction of the kernels $R_n(\mathbf{P},{\mathbf{p}}^{\prime },{\hat{\mathbf{k}}}^{\prime })$ . We will be interested only in large values of the principal quantum number n ≥ M ≫ 1, i.e., in the vicinity of the Schrodinger operator discrete Coulomb spectrum accumulation point relating to the subsystem with index 1. Here M is a certain large number.

Note that the Schrodinger operator continuous and discrete spectrum contributions of the pair subsystem (the first and second terms in the expression (34)) are orthogonal. Consequently, we can independently fix the unknown densities R(P, P′) and $R_n(\mathbf{P},{\mathbf{p}}^{\prime },{\hat{\mathbf{k}}}^{\prime })$ . The first one can be defined by the methods developed in the previous chapter. Our aim now is to describe the densities (weight functions) $R_n(\mathbf{P},{\mathbf{p}}^{\prime },{\hat{\mathbf{k}}}^{\prime })$ at large values of the main quantum number n. Mathematically it is connected to the spectral vicinity accumulation point contribution description of the pair subsystem discrete spectrum into the structure of the three-body Schrodinger operator continuous spectrum eigenfunction. It is the result that we are now aiming at.

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Metamerism and shade sorting

A.K. Roy Choudhury , in Principles of Colour and Appearance Measurement, 2015

5.11.4 Residual difference

As no improvement in the evaluation of special indices of metamerism was possible by spectral decomposition, due to near-identical fundamental stimuli of the respective members of most of the metameric pairs, efforts were made to propose improved general indices of metamerism on the basis of residual difference.

The respective samples of each metameric pair under study differ only in residuals, and very little in fundamentals. Therefore, a general index of metamerism, MI (RD) was proposed on the basis on the square-root of the sum of the squares of the residual difference, similar to the Bridgeman's index (Bridgeman and Hudson, 1969) based on reflectance differences.

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Rufus Ritchie, A Gentleman and A Scholar

Irina Yu. Sklyadneva , ... Evgenii V. Chulkov , in Advances in Quantum Chemistry, 2019

3.2.1 Spectral decomposition

To clarify the contribution of various phonon modes involved in the coupling, the spectral decomposition of λ _{k i} for some surface electron states was calculated. The calculated spectral representations λ _{k i} (ω) are shown in Figs. 5 and 6 as the average of the processes of emission and adsorption (see Eq. 1), which nearly coincide.

Fig. 5. Spectral representation of the coupling, λ _{k i} (ω), for four different surface states on Ag(110): (A) for the occupied SS at $\overline{\text{Y}}$ , (B) and (C) for the unoccupied SSs at points (B) $\overline{\text{X}}$ and (C) $\overline{\text{Y}}$ , and (D) for the SS marked in Fig. 1C as A. The shaded area represents the contribution of surface phonons located below the bulk modes (see Fig. 2, including the Rayleigh mode scattering).

Fig. 6. Spectral representation of the coupling, λ _{k i} (ω), for four different surface states on Cu(110): (A–C) the same as in Fig. 5, (D) for the SS marked in Fig. 1D as B.

The frequency distribution of λ _{k i} for the occupied surface state at $\overline{\text{Y}}$ (Fig. 5A) is unlike that for the Shockley state at the $\overline{\mathrm{\Gamma }}$ point on the (111) surface of Cu (see Fig. 1 in Ref. 40), despite the fact that the strength of e–ph interaction in both states is nearly the same. On the (111) surface, the Rayleigh mode makes a significant contribution: a peak in the low-energy part of the spectrum that dominates the spectral function. This mode with predominantly vertical vibrations of atoms actually determines the phonon-mediated intraband scattering in this surface state of s − p _z symmetry. ⁴⁸ λ _{k i} (ω) for the occupied surface state on the (110) surface (Figs. 5A and 6A) does not exhibit a very pronounced peak structure. In the low-energy part of the spectrum, the contribution of surface phonons located below the bulk modes gives only one noticeable peak, which is mainly determined by the coupling of electrons with in-plane vibrations of the first layer atoms in the direction perpendicular to the step edges (∼8   meV for Ag (110) and ∼12   meV for Cu (110)). In general, for the SS₃ band, the role of surface phonons in the e–ph interaction is not so important compared with the contribution of bulk-like vibrations, unlike the case of Shockley states on the (111) surface. This is expected because these surface states located at the edge of bulk electronic bands decay very slowly inside the film. The increasing overlap with final bulk electronic states results in a significant contribution to the scattering processes from bulk phonon modes.

An interesting feature of the spectrum is the contribution to λ _{k i} from the surface long-wave resonance (MS₇), which does not exist on the (111) surface. Its energy at the $\overline{\mathrm{\Gamma }}$ point is ∼19.5   meV for Cu(110) and ∼12.8   meV for Ag(110). This well-localized mode, characterized by vertical movements of atoms both at and under the step edges, actively participates in the e–ph coupling of all surface states. Especially notable is its contribution to λ _{k i} for the unoccupied surface state at $\overline{\text{Y}}$ (Figs. 5C and 6C). It should be noted that the surface gap mode in the vicinity of the $\overline{\text{Y}}$ point (mode S₅, see Fig. 2) with the in-plane transverse motion (perpendicular to the edges of the steps) of atoms in the upper layer and vertical displacements of atoms under the step edges also takes some part in the formation of this peak.

In contrast to the occupied surface state, λ _{k i} (ω) for the SS₂ state at $\overline{\text{Y}}$ has a number of dominant peaks associated with the contributions of surface phonons. This state, like the Shockley state on the (111) surface, is of s − p _z symmetry and is strongly localized just above the surface. The sharp peaks appearing in λ _{k i} (ω) at low energies (∼6–7   meV) are associated with the interband scattering involving surface modes split from bulk-like phonons at the SBZ boundary, including the Rayleigh mode (S1), which are mainly characterized by vertical vibrations of the first layer atoms.

The scattering provided by the vertical vibrations of atoms located under the step edges (MS₀) also makes a significant contribution, especially in the case of the Ag (110) surface, where this resonance is better localized in the subsurface layer. And, as mentioned above, the MS₇ long-wave resonances along with the surface gap mode in the vicinity of point $\overline{\text{Y}}$ (S₅) contribute significantly to the scattering of electrons. They predominate the intraband scattering, which varies by about 20% depending on the surface.

That the mechanisms of phonon-mediated scattering for an excited hole in the occupied surface state (SS₃) and for excited electrons in the unoccupied surface state (SS₂) are different, is due to their position in the band gap and, as a result, with the different phase spaces available for scattering processes. Thus, in comparison with the occupied surface states, the intraband contribution to the scattering increases, as well as the localized surface vibrations become more and more involved in the scattering processes with electrons. In addition, unlike the case of excited holes, a significant suppression of the high-energy part of the spectrum by e–ph matrix elements is observed. The growing participation of low- and middle-energy phonons in the scattering of excited electrons contributes to an increase in λ _{k i} .

This tendency persists for other unoccupied surface states and the contribution of surface phonons to λ _{k i} with increasing energy of excited electrons grows. Figs. 5D and 6C show the spectral decomposition of λ _{k i} for surface states marked A and B in the SS₂ band on Ag(110) and Cu(110), respectively. On moving away from $\overline{\text{Y}}$ , a noticeable increase in intraband transitions is observed, whose contribution reaches 45%. Since the intraband scattering is largely provided by low-energy surface phonons, the strength of the e–ph coupling also increases markedly. The intraband transitions are also very important in the e–ph scattering of the SS₄ surface state at $\overline{\text{X}}$ . Here, the intraband contribution to λ _{k i} dominates reaching ∼70%. And again, the involvement of low-energy surface phonons is decisive in the scattering of excited electrons (see Figs. 5D and 6D).

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Applications

Ali N. Akansu , Richard A. Haddad , in Multiresolution Signal Decomposition (Second Edition), 2001

One-Dimensional Subband Codec

Figure 7.3 displays the block diagram of a one-dimensional subband encoder/decoder or codec. The input signal x(n ) goes through a spectral decomposition via an analysis filter bank. The subbands of the analysis filter bank should be properly designed to match the shape of the input spectrum. This is a very important point that significantly affects performance of the system. Compression bits are then allocated to the subband signals based on their spectral energies. These allocated bits are used by quantizers. An entropy encoder follows the quantizers to remove any remaining redundancy. The compressed bit stream { b_i } is transmitted through a communication channel or stored in a storage medium. We assume an ideal channel or storage medium in this example. Similarly, entropy decoding, inverse quantization, and synthesis filtering operations are performed at the receiver in order to obtain the decompressed signal $\hat{x}(n)$ . In reality, a communications channel introduces some bit errors during transmissions that degrade the quality of the synthesized signal at the decoder (receiver).

Figure 7.3. The block diagram of a subband codec.

The blocks in the subband codec system (Fig. 7.3) are briefly described as follows:

Analysis Filterbank

Hierarchical filter banks are used in most coding applications. The subband tree structure which defines the spectral decomposition of the input signal should match input spectrum. Additionally, several time- and frequency-domain tools were introduced in Chapter 4 for optimal filter bank design. The implementation issues along with the points made here will yield practical solutions.

Quantization

Lossy compression techniques require an efficient entropy reduction scheme. A quantizer is basically a bit compressor. It reduces the bit rate and introduces irreversable quantization noise. Hence, it is called lossy compression.

Entropy Encoder

The quantizer generates an output with some redundancy. Any entropy encoder, such as the Huffmann coder, exploits this redundancy. Note that the entropy encoder encodes the source in a lossless fashion that is perfectly reversible. The output bit stream of the entropy encoder is compressed and ready for transmission or storage.

Channel or Storage Medium

The capacity of a communications channel or storage medium at a given bit error rate is the defining factor. The encoder aims to achieve the necessary compression rate in order to fit the original source data into the available channel or storage capacity. Note that wireline (e.g., telephone lines) and wireless (e.g., cell phones) channels have different physical media and engineering properties that are handled accordingly.

Similarly, entropy decoders, inverse quantizers, and synthesis filter banks perform inverse operations at the receiver or decoder.

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Oil reservoir quality assisted by machine learning and evolutionary computation

M.C. Kuroda , ... J.P. Papa , in Bio-Inspired Computation and Applications in Image Processing, 2016

5.3 Prediction of porosity into the 3D grid

As explained, in order to predict porosity at the seismic scale, GA was used to select the best set of attributes among several options described in Table 13.1 , using the results of MLP as selection criterion. The best GA training results selected the seismic attributes "spectral decomposition of 40 Hz" and "phase rotation" as the best variables to predict porosity, with errors of 0.0648 and 0.0649, respectively. This same set of attributes was chosen to predict lithology, as shown previously. Such choice is considered appropriate, mainly because these seismic attributes are often used for porosity prediction ( Chopra and Marfurt, 2007).

Fig. 13.13 shows the set of seismic attributes chosen by the GA for the task of electrofacies prediction, well cutting, and porosity prediction. Although AI was not chosen, it was used because of its relevance in the description of geological features, as mentioned before (Leiphart and Hart, 2001; Avadhani et al., 2006; Chopra and Marfurt, 2007).

Figure 13.13. Selected seismic attributes by GA for the prediction of porosity (PHIE), drill cutting, and facies.

Phase rotation and acoustic impedance were the attributes more frequently chosen by the algorithm.

Regarding the task of porosity prediction by MLP using as input the AI seismic attribute, the best MLP training algorithm was the Levenberg–Marquardt backpropagation (Fig. 13.14). After defining the training function, other tests were conducted to identify the best number of neurons (Fig. 13.15). The training steps with AI and phase rotation were the ones that yielded more error variations, considering the number of neurons. Moreover, the combination of all variables led to the smallest errors in both tests. The results of porosity extrapolation are illustrated in Fig. 13.16. Indeed, the MLP training with the combination of all variables was closer to the original values than the other results.

Figure 13.14. MSE error, considering different training function for porosity extrapolation, using spectral decomposition of 40 Hz (DE 40), acoustic impedance (AI), phase rotation (PHASE), and all variables.

Figure 13.15. MSE error considering a different number of neurons for porosity extrapolation using spectral decomposition of 40 Hz (DE 40), acoustic impedance (AI), phase rotation (PHASE), and all variables.

Figure 13.16. Comparison of the results concerning MLP porosity prediction by spectral decomposition of 40 Hz (DE 40), acoustic impedance (AI), both variables in graph (a); phase rotation (PHASE), and all variables in graph (b).

5.3.1 3D porosity prediction by GRNN

The GRNN algorithm was also employed to predict porosity, using parameters defined in Section 4.4 . The training of spectral decomposition of 40 Hz and phase rotation attributes (with MSE errors of 0.013) achieved constant values of porosity. However, better results were obtained using AI and the combination of all variables (with MSE error of 0.0141 and 0.0076, respectively), although the first variable showed low similarity with the original porosity values (Fig. 13.17).

Figure 13.17. Comparison of the results of GRNN porosity prediction by spectral decomposition of 40 Hz (DE 40), acoustic impedance (AI), in graph (a); phase rotation (PHASE), and all variables in graph (b).

The porosity prediction by MLP and GRNN using all variables (spectral decomposition of 40 Hz, AI and phase rotation) resulted in similar values. In order to compare these results, the crossplot (Fig. 13.18) shows that the MLP prediction was a little better, with a 65% correlation. Therefore, this method was chosen to predict porosity in the field.

Figure 13.18. Crossplot of the estimated porosity calculated by GRNN and MLP algorithms, using all seismic attributes.

The MLP prediction was a little better, with a 65% correlation.

The distribution of porosity (Fig. 13.19) showed similar results compared to the seismofacies distribution, yielding higher values at the top of the study area.

Figure 13.19. Details of the seismic amplitude (a) and the distribution of porosity in section AA′ (b). The higher values of porosity at the top of the reservoir are associated with the good and regular sandstone previously predicted. The gamma ray well log is highlighted for well W_1.

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SPECTROSCOPY | Fourier Transform Spectroscopy

T. Fromherz , in Encyclopedia of Modern Optics, 2005

Advantage of FTS Over Dispersive Spectrometer

The two main advantages of FTS over dispersive spectroscopic methods, using, for example, a grating for the spectral decomposition of an incoming radiation, are directly connected with the working principle of FT spectrometers; high optical throughput and multiplexed measurement over the whole spectral range of interest. These advantages are discussed in the following, in more detail.

For a fair comparison of the optical throughput, spectrometers with equal spectral resolving power ρ have to be compared. Here, ρ = σ _max/Δσ, where σ _max is the maximum wavenumber analyzed and Δσ is the spectral resolution. For an ideal Fourier transform spectrometer, the limit for σ _max depends on the divergence angle of the beams propagating in the spectrometer. As sketched in Figure 5, the divergence angle α is determined by the radius r _a of the entrance aperture and the focal length f of the collimating lens by α ≈ r _a/f. This angle causes a phase shift between the rays propagating parallel to and those propagating under the angle of α inclined to the optical axis through the interferometer. The phase difference φ _div depends on the mirror displacement according to ${\phi }_{\text{div}}=4\pi \sigma \text{Δ}x((1/cos(\alpha ))-1)$ and can be approximated for small α by φ _div ≈ 2πσΔxα ². It can be shown, that as long as φ _div does not exceed approximately π, the influence of the divergence on the interferogram can be tolerated, i.e., σ _max ≈ 1/(2Δx _max α ²). Using Δσ ≈ 1/(2Δx _max) for the resolution, one obtains for the resolving power of a FT spectrometer, ρ = (f/r _a)².

Figure 5. Entrance aperture and collimating element for an interferometer. For simplicity, the beamsplitter and fixed mirror are omitted in the sketch. The extreme rays entering the interferometer are indicated by the full and broken lines. Due to divergence α, a monochromatic light with wavenumber σ ₀ produces all periods in the range [4πσ ₀,4πσ ₀/cos(α)] in the interferogram.

The optical throughput Θ describes the allowable energy per unit time that the system can let through and is related to area $\pi r_{\text{a}}^2$ of the aperture and the solid angle Ω of the collimating (or focusing) optics by $\mathbf{\Theta }=\pi r_{\text{a}}^2\mathbf{\Omega }$ . Using Ω = A _FTS/f ², where A _FTS is the clear area of the collimating lens (for a collimating mirror, A _FTS is the area of the mirror projected on a plane perpendicular to the beam direction), results in ${\mathbf{\Theta }}_{\text{FTS}}=\pi A_{\text{FTS}}{r}_{\text{a}}^2/f^2=\pi A_{\text{FTS}}/\rho$ . For a grating spectrometer, ${\mathbf{\Theta }}_{\text{grating}}=h{A}_{\text{G}}/f\rho .$ (see respective chapter of this encyclopedia) where h is the height of the slit, f the focal length of the collimating optics, and A _G the projected area of the grating. Generally, h/f is smaller than 1/20 in up-to-date grating spectrometers. Thus, for the same resolving power ρ and similar instrument size, FT spectrometer can offer a more than πf/h ≈ 60 times larger energy gathering capability. Therefore, for measurements with weak signals in that the detector noise is the dominant noise source, the large throughput of a FTS greatly increases the spectral signal to noise ratio that can be achieved for a fixed measuring time.

The other major principal advantage compared to dispersive methods relies on the simultaneous (multiplexed) gathering of information about all spectral bands Δσ over a broad spectral range from σ _min and σ _max. For a grating spectrometer using a single detector element, the measurement time T _m available for recording the spectrum in that range has to be distributed between the $M=({\sigma }_{\text{max}}-{\sigma }_{\text{min}})/\text{Δ}\sigma$ spectral bands so that for a single spectral band the available integration time is T _m/M and the integrated signal increases proportional to T _m/M If the dominant noise source is the detector noise which is independent of the signal level, then the noise will be proportional to (T _m/M)^1/2 and, therefore, the signal to noise ratio for the grating spectrometer (S/N)_G is:

[19] ${(S/N)}_{\text{G}}\propto {(T_{\text{m}}/M)}^{1/2}$

For a FTS the situation is different, since it detects in the band [σ _min,σ _max] all M small bands over the whole measurement time T _m via their contribution to the interferogram. So the integrated signal in a small band Δσ is proportional to T _m. If the noise is again assumed to be random and independent of the signal level, for the FTS:

[20] ${(S/N)}_{\text{FTS}}\propto T_{\text{m}}^{1/2}$

results. For the same detector, assumed in the grating spectrometer and in the FTS, the proportionality constants are the same in eqns [19] and [20] and, therefore

[21] $\frac{{(S/N)}_{\text{FTS}}}{{(S/N)}_{\text{G}}}=M^{1/2}$

Equation [21] shows that the multiplex advantage becomes increasingly important for measurements of broad spectral bands with high resolution. In this case, an enhancement of the (S/N)_FTS over (S/N)_G by 2–3 orders of magnitude can be achieved. However, it has to be noted that the multiplex advantage of a FTS can be exploited only in cases where the detector noise is the dominant source of noise. In the visible and near infrared spectral region, low noise detectors, that allow single photon detection, are available. With these detectors, the spectral noise is dominated by the photon noise that is proportional to the square root of the intensity. It is evident that in this case the multiplexed detection is not an advantage, since for M spectral bands of width Δσ contributing to the interferogram at a given mirror position, the noise is enhanced by M ^1/2 compared to the noise that would be measured for a single band (here, it is assumed that all M bands contribute with the same intensity to the interferogram). Consequently, in this case, the noise enhancement by M ^1/2 cancels the beneficial factor M ^1/2 in eqn [21] and with single-detector grating and FT spectrometers, the same S/N ratio is achieved for a given measurement time T _m. In addition, using linear detector arrays with dispersive spectrometers, the multiplex advantage increases the S/N ratio for the dispersive spectrometer. However, detector arrays and low-noise detectors are mainly available for spectral regions with σ > 6000 cm⁻¹, i.e., in the near-infrared and visible spectral region. In the FIR and MIR region (σ < 6000 cm⁻¹) in that such detectors are absent, the multiplexing inherent in FTS makes it the method of choice if low-noise, broadband, and high-resolution spectroscopic data are required.

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https://www.sciencedirect.com/science/article/pii/B0123693950008393

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Source: https://www.sciencedirect.com/topics/engineering/spectral-decomposition

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