\r\n\tIn particular, this book presents topics related to Audio Signal Processing based on the different perspectives of the following: pattern recognition on audio, audio processing, forensic audio, digital filtering, and frequency analysis, and digital signal processing chip for audio, although other topics can be included, too. The most innovative advances on Audio Signal Processing will be included in this book, in order to show the reader, the new researched and developed approaches.
\r\n
\r\n\tSpecific cases of voice applications are welcome, where the Voice over IP (VoIP), internet of things (IoT), deep learning (DL) approaches, etc., are very useful including the recent technologies applied on voice and audio.
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Travieso-Gonzalez",publishedDate:null,coverURL:"https://cdn.intechopen.com/books/images_new/8213.jpg",keywords:"Pattern Recognition, Audio Identification, Audio Processing Algorithm, Audio Enhancer, Human Voice Patterns, Text to Speech, Forensic Audio Enhancement, Audio Evidence, Filtering Audio, Wavelet Analysis, Microprocessor for Audio, DSP for Audio",numberOfDownloads:null,numberOfWosCitations:0,numberOfCrossrefCitations:0,numberOfDimensionsCitations:null,numberOfTotalCitations:null,isAvailableForWebshopOrdering:!0,dateEndFirstStepPublish:"October 24th 2019",dateEndSecondStepPublish:"March 2nd 2020",dateEndThirdStepPublish:"May 1st 2020",dateEndFourthStepPublish:"July 20th 2020",dateEndFifthStepPublish:"September 18th 2020",remainingDaysToSecondStep:"a year",secondStepPassed:!0,currentStepOfPublishingProcess:5,editedByType:null,kuFlag:!1,biosketch:null,coeditorOneBiosketch:null,coeditorTwoBiosketch:null,coeditorThreeBiosketch:null,coeditorFourBiosketch:null,coeditorFiveBiosketch:null,editors:[{id:"27170",title:"Prof.",name:"Carlos",middleName:"M.",surname:"Travieso-Gonzalez",slug:"carlos-travieso-gonzalez",fullName:"Carlos Travieso-Gonzalez",profilePictureURL:"https://mts.intechopen.com/storage/users/27170/images/system/27170.jpeg",biography:"Carlos M. Travieso-González received his M.Sc. degree in 1997 in Telecommunication Engineering at the Polytechnic University of Catalonia (UPC), Spain; and his Ph.D. degree in 2002 at the University of Las Palmas de Gran Canaria (ULPGC-Spain). He is a Full Professor in Signal Processing and Pattern Recognition and Head of the Signals and Communications Department at ULPGC; teaching from 2001 in subjects on signal processing and learning theory. His research lines are biometrics, biomedical signals and images, data mining, classification systems, signal and image processing, machine learning, and environmental intelligence. He has taken part in 51 international and Spanish research projects, some of them as head researcher. He is co-author of 4 books, co-editor of 24 proceedings books, guest editor for 8 JCR-ISI international journals and up to 24 book chapters. He has had over 440 papers published in international journals and conferences (74 of them indexed on JCR – ISI - Web of Science). He has published 7 patents with the Spanish Patent and Trademark Office. He has been supervisor on 8 PhD theses (12 more are under supervision), and 130 Master theses. He is the founder of The IEEE IWOBI Conference series (and President of its steering committee), The InnoEducaTIC conference series, and The APPIS conference series. He is evaluator of project proposals for the European Union (H2020), Medical Research Council (MRC – UK), Spanish Government (ANECA - Spain), Research National Agency (ANR - France), DAAD (Germany), Argentinian Government, and Colombian Institutions. He has been reviewer in different indexed international journals (<70) and conferences (<220) since 2001. He is a member of IASTED Technical Committee on Image Processing from 2007 and member of IASTED Technical Committee on Artificial Intelligence and Expert Systems from 2011. He will be ACM-APPIS 2021 General Chair and IEEE-IWOBI 2020 and 2020, and was ACM-APPIS 2020 General Chair, IEEE-IWOBI 2019, General Chair APPIS 2019 General Chair, IEEE-IWOBI 2018 General Chair, APPIS 2018 General Chair, InnoEducaTIC 2017 General Chair, IEEE-IWOBI 2017 General Chair, IEEE-IWOBI 2015 General Chair, InnoEducaTIC 2014 General Chair, IEEE-IWOBI 2014 General Chair, IEEE-INES 2013 General Chair, NoLISP 2011 General Chair, JRBP 2012 General Chair and IEEE-ICCST 2005 Co-Chair. He is Associate Editor for the Computational Intelligence and Neuroscience Journal (Hindawi – Q2 JCR-ISI). He was Vice-Dean from 2004 to 2010 at the Higher Technical School of Telecommunication Engineers in ULPGC; and Vice-Dean of Graduate and Postgraduate Studies from March 2013 to November 2017. He won “Catedra Telefonica” Awards in Modality of Knowledge Transfer, in the editions 2017, 2018 and 2019.",institutionString:"University of Las Palmas de Gran Canaria",position:null,outsideEditionCount:0,totalCites:0,totalAuthoredChapters:"6",totalChapterViews:"0",totalEditedBooks:"2",institution:{name:"University of Las Palmas de Gran Canaria",institutionURL:null,country:{name:"Spain"}}}],coeditorOne:null,coeditorTwo:null,coeditorThree:null,coeditorFour:null,coeditorFive:null,topics:[{id:"24",title:"Technology",slug:"technology"}],chapters:null,productType:{id:"1",title:"Edited Volume",chapterContentType:"chapter",authoredCaption:"Edited by"},personalPublishingAssistant:{id:"247865",firstName:"Jasna",lastName:"Bozic",middleName:null,title:"Ms.",imageUrl:"https://mts.intechopen.com/storage/users/247865/images/7225_n.jpg",email:"jasna.b@intechopen.com",biography:"As an Author Service Manager, my responsibilities include monitoring and facilitating all publishing activities for authors and editors. From chapter submission and review to approval and revision, copyediting and design, until final publication, I work closely with authors and editors to ensure a simple and easy publishing process. I maintain constant and effective communication with authors, editors and reviewers, which allows for a level of personal support that enables contributors to fully commit and concentrate on the chapters they are writing, editing, or reviewing. I assist authors in the preparation of their full chapter submissions and track important deadlines and ensure they are met. I help to coordinate internal processes such as linguistic review, and monitor the technical aspects of the process. As an ASM I am also involved in the acquisition of editors. Whether that be identifying an exceptional author and proposing an editorship collaboration, or contacting researchers who would like the opportunity to work with IntechOpen, I establish and help manage author and editor acquisition and contact."}},relatedBooks:[{type:"book",id:"6126",title:"Colorimetry and Image Processing",subtitle:null,isOpenForSubmission:!1,hash:"f74525de04361957bd947a45b0e64378",slug:"colorimetry-and-image-processing",bookSignature:"Carlos M. 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\n
1. Introduction
\n
Over the past several decades, there has been a large research interest in the problem of single-channel sound source separation. Such work focuses on the task of separating a single mixture recording into its respective sources and is motivated by the fact that real-world sounds are inherently constructed by many individual sounds (e.g., human speakers, musical instruments, background noise, etc.). While source separation is difficult, the topic is highly motivated by many outstanding problems in audio signal processing and machine learning, including the following:
Speech denoising and enhancement—the task of removing background noise (e.g., wind, babble, etc.) from recorded speech and improving speech intelligibility for human listeners and/or automatic speech recognizers
Content-based analysis and processing—the task of extracting and/or processing audio based on semantic properties of the recording such as tempo, rhythm, and/or pitch
Music transcription—the task of notating an audio recording into a musical representation such as a musical score, guitar tablature, or other symbolic notations
Audio-based forensics—the task of examining, comparing, and evaluating audio recordings for scientific and/or legal matters
Audio restoration—the task of removing imperfections such as noise, hiss, pops, and crackles from (typically old) audio recordings
Music remixing and content creation—the task of creating a new musical work by manipulating the content of one or more previously existing recordings
\n
\n
\n
2. Nonnegative matrix factorization
\n
\n
2.1 NMF model
\n
Nonnegative matrix factorization is a process that approximates a single nonnegative matrix as the product of two nonnegative matrices. It is defined by
\n
\n\nV\n≈\nWH\n\nE1
\n
\n\n\nV\n∈\n\nR\n+\n\n\nN\nf\n\n×\n\nN\nt\n\n\n\n\n is a nonnegative input matrix. \n\nW\n∈\n\nR\n+\n\n\nN\nf\n\n×\n\nN\nz\n\n\n\n\n is a matrix of basis vectors, basis functions, or dictionary elements; \n\nH\n∈\n\nR\n+\n\n\nN\nz\n\n×\n\nN\nt\n\n\n\n\n is a matrix of corresponding activations, weights, or gains; and \n\n\nN\nf\n\n\n is the number of rows of the input matrix. \n\n\nN\nt\n\n\n is the number of columns of the input matrix; \n\n\nN\nz\n\n\n is the number of basis vectors [1].
\n
\n\n\nV\n∈\n\nR\n+\n\n\nN\nf\n\n×\n\nN\nt\n\n\n\n\n—original nonnegative input data matrix
Each column is an \n\n\nN\nf\n\n\n -dimensional data sample.
Each row represents a data feature.
\n
\n\n\nW\n∈\n\nR\n+\n\n\nN\nf\n\n×\n\nN\nz\n\n\n\n\n\n—matrix of basis vectors, basis functions, or dictionary elements.
A column represents a basis vector, basis function, or dictionary element.
Each column is not orthonormal, but commonly normalized to one.
\n
\n\n\nH\n∈\n\nR\n+\n\n\nN\nz\n\n×\n\nN\nt\n\n\n\n\n —matrix of activations, weights, encodings, or gains.
A row represents the gain of a corresponding basis vector.
Each row is not orthonormal, but sometimes normalized to one.
\n
When used for audio applications, NMF is typically used to model spectrogram data or the magnitude of STFT data [2]. That is, we take a single-channel recording, transform it into the time-frequency domain using the STFT, take the magnitude or power V, and then approximate the result as \n\nV\n≈\nWH\n\n. In doing so, NMF approximates spectrogram data as a linear combination of prototypical frequencies or spectra (i.e., basis vectors) over time.
\n
This process can be seen in Figure 1 [3], where a two-measure piano passage of “Mary Had a Little Lamb” is shown alongside a spectrogram and an NMF factorization. Notice how W captures the harmonic content of the three pitches of the passage and H captures the time onsets and gains of the individual notes. Also note that \n\n\nN\nz\n\n\n is typically chosen manually or using a model selection procedure such as cross-validation and \n\n\nN\nf\n\n\n and \n\n\nN\nt\n\n\n are a function of the overall recording length and STFT parameters (transform length, zero-padding size, and hop size).
\n
Figure 1.
NMF of a piano performing “Mary had a little lamb” for two measures with \n\n\nN\nz\n\n\n = 3. Notice how matrix W captures the harmonic content of the three pitches of the passage and matrix H captures the time onsets and gains of the individual notes [3].
\n
This leads to two related interpretations of how NMF models spectrogram data. The first interpretation is that the columns of V (i.e., short-time segments of the mixture signal) are approximated as a weighted sum of basis vectors as shown in Figure 2 and Eq. (2):
NMF interpretation I. the columns of V (i.e., short-time segments of the mixture signal) are approximated as a weighted sum or mixture of basis vectors W [3].
\n
The second interpretation is that the entire matrix V is approximated as a sum of matrix “layers,” as shown in Figure 3 and Eq. (3).
NMF interpretation II. The matrix V (i.e., the mixture signal) is approximated as a sum of matrix “layers” [3].
\n
The application of NMF on noisy speech can be seen in Figure 4.
\n
Figure 4.
Applying NMF on noisy speech.
\n
\n
\n
2.2 Optimization formulation
\n
To estimate the basis matrix W and the activation matrix H for a given input data matrix V, NMF algorithm is formulated as an optimization problem. This is written as:
where \n\nD\n\n\nV\n\nWH\n\n\n\n\n is an appropriately defined cost function between V and W H and the inequalities \n\n≥\n\n are element-wise. It is also common to add additional equality constraints to require the columns of W to sum to one, which we enforce. When \n\nD\n\n\nV\n\nWH\n\n\n\n\n is additively separable, the cost function can be reduced to
where \n\n\n\n\n\nft\n\n\n indicates its argument at row f and column t and \n\nD\n\n\nV\n\nWH\n\n\n\n\n is a scalar cost function measured between V and WH.
\n
Popular cost functions include the Euclidean distance metric, Kullback-Liebler (KL) divergence, and Itakura-Saito (IS) divergence. Both the KL and IS divergences have been found to be well suited for audio purposes. In this work, we focus on the case where \n\nd\n\n\nq\n\np\n\n\n\n\n is generalized (non-normalized) KL divergence:
where \n\n\n\n\n\nft\n\n\n indicates its argument at row f and column t and \n\nd\n\n\nq\n\np\n\n\n\n\n is a scalar cost function measured between q and p.
\n
This results in the following optimization formulation:
Given this formulation, we notice that the problem is not convex in W and H, limiting our ability to find a globally optimal solution to Eq. (7). It is, however, biconvex or independently convex in W for a fixed value of H and convex in H for a fixed value of W, motivating the use of iterative numerical methods to estimate locally optimal values of W and H.
\n
\n
\n
2.3 Parameter estimation
\n
To solve Eq. (7), we must use an iterative numerical optimization technique and hope to find a locally optimal solution. Gradient descent methods are the most common and straightforward for this purpose but typically are slow to converge. Other methods such as Newton’s method, interior-point methods, conjugate gradient methods, and similar [4] can converge faster but are typically much more complicated to implement, motivating alternative approaches.
\n
The most popular alternative that has been proposed is by Lee and Seung [1, 5] and consists of a fast, simple, and efficient multiplicative gradient descent-based optimization procedure. The method works by breaking down the larger optimization problem into two subproblems and iteratively optimizes over W and then H, back and forth, given an initial feasible solution. The approach monotonically decreases the optimization objective for both the KL divergence and Euclidean cost functions and converges to a local stationary point.
\n
The approach is justified using the machinery of majorization-minimization (MM) algorithms [6]. MM algorithms are closely related to expectation maximization (EM) algorithms. In general, MM algorithms operate by approximating an optimization objective with a lower bound auxiliary function. The lower bound is then maximized instead of the original function, which is usually more difficult to optimize.
\n
Algorithm 1 shows the complete iterative numerical optimization procedure applied to Eq. (7) with the KL divergence, where the division is element-wise,\n\n⊗\n\nis an element-wise multiplication, and 1 is a vector or matrix of ones with appropriately defined dimensions [5].
\n
Algorithm 1 KL-NMF parameter estimation
\n
Procedure KL-NMF (\n\nV\n∈\n\nR\n+\n\n\nN\nf\n\n×\n\nN\nt\n\n\n\n\n//input data matrix.
NMF is an optimization technique using EM algorithm in terms of matrix, whereas probabilistic latent component analysis (PLCA) is also an optimization technique using EM algorithm in terms of probability. In PLCA, we are going to incorporate probabilities of time and frequency. In the next section, the development of PLCA-based algorithm to incorporate time-frequency constraints is discussed.
\n
\n
\n
\n
3. A probabilistic latent variable model with time-frequency constraints
\n
Considering this approach, we now develop a new PLCA-based algorithm to incorporate the time-frequency user-annotations. For clarity, we restate the form of the symmetric two-dimensional PLCA model we use:
Compared to a modified NMF formulation, incorporating optimization constraints as a function of time, frequency, and sound source into the factorized PLCA model is particularly interesting and motivating to our focus.
\n
Incorporating prior information into this model, and PLCA in general, can be done in several ways. The most commonly used methods are by direct observations (i.e., setting probabilities to zero, one, etc.) or by incorporating Bayesian prior probabilities on model parameters. Direct observations do not give us enough control, so we consider incorporating Bayesian prior probabilities. For the case of Eq. (10), this would result in independently modifying the factor terms \n\np\n\n\nf\n\nz\n\n\n\n\n, \n\np\n\n\nt\n\nz\n\n\n\n\n, or \n\np\n\nz\n\n\n. Common prior probability distributions used for this purpose include Dirichlet priors [7], gamma priors [8], and others.
\n
Given that we would like to incorporate the user-annotations as a function of time, frequency, and sound source, however, we notice that this is not easily accomplished using standard priors. This is because the model is factorized, and each factor is only a function of one variable and (possibly) conditioned by another, making it difficult to construct a set of prior probabilities that, when jointly applied to \n\np\n\n\nf\n\nz\n\n\n\n\n, \n\np\n\n\nt\n\nz\n\n\n\n\n, and/or \n\np\n\nz\n\n\n, would encourage or discourage one source or another to explain a given time-frequency point. We can see this more clearly when we consider PLCA to be the following simplified estimation problem:
where \n\nX\n\nf\nt\n\n\n is the observed data that we model as the product of three distinct functions or factors \n\nφ\n\nz\n\n\n, \n\nφ\n\nf\nz\n\n\n, and \n\nφ\n\nt\nz\n\n\n. Note, each factor has different input arguments and each factor has different parameters that we wish to estimate via EM. Also, forget for the moment that the factors must be normalized probabilities.
\n
Given this model, if we wish to incorporate additional information, we could independently modify:
\n\n\nφ\n\nz\n\n\n to incorporate past knowledge of the variable z
\n\n\nφ\n\nf\nz\n\n\n to incorporate past knowledge of the variable f and z
\n\n\nφ\n\nt\nz\n\n\n to incorporate past knowledge of the variable t and z
\n
This way of manipulation allows us to maintain our factorized form and can be thought of as prior-based regularization. If we would like to incorporate additional information/regularization that is a function of all three variables z, f, and t, then we must do something else. The first option would be to try to simultaneously modify all factors together to impose regularization that is a function of all three variables. This is unfortunately very difficult—both conceptually difficult to construct and practically difficult to algorithmically solve.
\n
This motivates the use of posterior regularization (PR). PR provides us with an algorithmic mechanism via EM to incorporate constraints that are complementary to prior-based regularization. Instead of modifying the individual factors of our model as we saw before, we directly modify the posterior distribution of our model. The posterior distribution of our model, very loosely speaking, is a function of all random variables of our model. It is natively computed within each E step of EM and is required to iteratively improve the estimates of our model parameters. In this example, the posterior distribution would be akin to \n\nφ\n\nz\nf\nt\n\n\n, which is a function of t, f, and z, as required. We now formally discuss PR below, beginning with a general discussion and concluding with the specific form of PR we employ within our approach.
\n
\n
3.1 Posterior regularization
\n
The framework of posterior regularization, first introduced by Graca, Ganchev, and Taskar [9, 10], is a relatively new mechanism for injecting rich, typically data-dependent constraints into latent variable models using the EM algorithm. In contrast to standard Bayesian prior-based regularization, which applies constraints to the model parameters of a latent variable model in the maximization step of EM, posterior regularization applies constraints to the posterior distribution (distribution over the latent variables, conditioned on everything else) computation in the expectation step of EM. The method has found success in many natural language processing tasks, such as statistical word alignment, part-of-speech tagging, and similar tasks that involve latent variable models.
\n
In this case, what we do is constrain the distribution q in some way when we maximize the auxiliary bound \n\nF\n\nq\nΘ\n\n\n with respect to q in the expectation step of an EM algorithm, resulting in
where \n\nΩ\n\nq\n\n\n constrains the possible space of q.
\n
Note, the only difference between Eq. (12) and our past discussion on EM is the added term \n\nΩ\n\nq\n\n\n. If \n\nΩ\n\nq\n\n\n is set to zero, we get back the original formulation and easily solve the optimization by setting q = p without any computation (except computing the posterior p). Also note to denote the use of constraints in this context, the term “weakly supervised” was introduced by Graca [11] and is similarly adopted here.
\n
This method of regularization is in contrast to prior-based regularization, where the modified maximization step would be
where \n\nΩ\n\nΘ\n\n\n constrains the model parameter \n\nΘ\n\n.
\n
\n
\n
3.2 Linear grouping expectation constraints
\n
Given the general framework of posterior regularization, we need to define a meaningful penalty \n\nΩ\n\nq\n\n\n for which we map our annotations. We do this by mapping the annotation matrices to linear grouping constraints on the latent variable z. To do so, we first notice that Eq. (12) decouples for each time-frequency point for our specific model. Because of this, we can independently solve Eq. (12) for each time-frequency point, making the optimization much simpler. When we rewrite our E step optimization using vector notation, we get
subject to \n\nq\nft\nT\n\n1\n=\n1\n,\n\nq\nft\n\n≥\n0\n\nE14
\n
where q and \n\np\n\n\nz\n\nf\nt\n\n\n\n\n for a given value of f and t is written as \n\n\nq\nft\n\n\n and \n\n\np\nft\n\n\n without any modification; we note q is optimal when equal to \n\np\n\n\nz\n\nf\nt\n\n\n\n\n as before.
\n
We then apply our linear grouping constraints independently for each time-frequency point:
Subject to \n\nq\nft\nT\n\n1\n=\n1\n,\n\nq\nft\n\n≥\n0\n,\n\nE15
\n
where we define \n\n\nλ\nft\n\n=\n\n\n\n\nΛ\n\nft\n1\n\n\n.\n.\n…\n…\n\nΛ\n\nft\n1\n\n\n\nΛ\n\nft\n2\n\n\n…\n…\n…\n\nΛ\n\nft\n2\n\n\n\n\nT\n\n∈\n\nR\n\nN\nz\n\n\n\n as the vector of user-defined penalty weights, \n\nT\n\n is a matrix transpose, \n\n≥\n\n is element-wise greater than or equal to, and 1 is a column vector of ones. In this case, positive-valued penalties are used to decrease the probability of a given source, while negative-valued coefficients are used to increase the probability of a given source. Note the penalty weights imposed on the group of values of z that correspond to a given source s are identical, linear with respect to the z variables, and applied in the E step of EM, hence the name “linear grouping expectation constraints.”
\n
To solve the above optimization problem for a given time-frequency point, we form the Lagrangian
where exp{} is an element-wise exponential function.
\n
Notice the result is computed in closed form and does not require any iterative optimization scheme as may be required in the general posterior regularization framework [9], minimizing the computational cost when incorporating the constraints. Also note, however, that this optimization must be solved for each time-frequency point of our spectrogram data for each E step iteration of our final EM parameter estimation algorithm.
\n
\n
\n
3.3 Parameter estimation
\n
Now knowing the posterior-regularized expectation step optimization, we can derive a complete EM algorithm for a posterior-regularized two-dimensional PLCA model (PR-PLCA):
where \n\n\nΛ\n¯\n\n=\nexp\n\n\n−\nΛ\n\n\n\n. The entire algorithm is outlined in Algorithm 2. Notice we continue to maintain closed-form E and M steps, allowing us to optimize further and draw connections to multiplicative nonnegative matrix factorization algorithms.
\n
Algorithm 2 PR-PLCA with linear grouping expectation constraints
\n
Procedure PLCA (
\n
\n\n\nV\n∈\n\nR\n+\n\n\nN\nf\n\n×\n\nN\nt\n\n\n\n\n//observed normalized data
which fully specifies the iterative updates. By putting Eqs. (30) and (31) in matrix notation, we specify the multiplicative form of the proposed method in Algorithm 3.
\n
Algorithm 3. PR-PLCA with linear grouping expectation constraints in matrix notation
\n
Procedure PLCA (
\n
\n\n\nV\n∈\n\nR\n+\n\n\nN\nf\n\n×\n\nN\nt\n\n\n\n\n//observed normalized data
4. An iterative posterior NMF method for speech enhancement in the presence of additive Gaussian noise (proposed algorithm)
\n
Over the past several years, research has been carried out in single-channel sound source separation methods. This problem is motivated by speech denoising, speech enhancement [13], music transcription [14], audio-based forensic, and music remixing. One of the most effective approach is nonnegative matrix factorization (NMF) [5]. The user-annotations can be used to obtain the PR terms [15]. If the number of sources is more, then it is difficult to identify sources in the spectrogram. In such cases, the user interaction-based constraint approaches are inefficient.
\n
In order to avoid the previous problem, in the proposed method, an automatic iterative procedure is introduced. The spectral components of speech and noise are modeled as Gamma and Rayleigh, respectively [16].
\n
\n
4.1 Notation and basic concepts
\n
Let noisy speech signal x[n] be the sum of clean speech s[n] and noise d[n] and their corresponding magnitude spectrogram be represented as
where f represents the frequency bin and t the frame number. The observed magnitudes in time-frequency are arranged in a matrix X\n\n∈\n\nR\n+\n\nf\n×\nt\n\n\n\n of nonnegative elements. The source separation algorithms based on NMF pursue the factorization of X as a product of two nonnegative matrices, \n\nW\n=\n\n\nw\n1\n\n\nw\n2\n\n…\n\nw\nR\n\n\n∈\n\nR\n+\n\nf\n×\nR\n\n\n\n in which the columns collect the basis vectors and \n\nH\n=\n\n\n\nh\n1\nT\n\n\nh\n2\nT\n\n\n.\n…\n\n\nh\nR\nT\n\n\nT\n\n∈\n\nR\n+\n\nR\n×\nt\n\n\n\n that collects their respective weights, i.e.,
There are several ways to incorporate the user-annotations into latent variable models, for instance, by using the suitable regularization functions. For expectation maximization (EM) algorithms, posterior regularization was introduced by [9, 11]. This method is data dependent. This method gives richness and also gives the constraints on the posterior distributions of latent variable models. The applications of this method is used in many natural language processing tasks like statistical word alignment, part-of-speech tagging. The main idea is to constrain on the distribution of posterior, when computing expectation step in EM algorithm.
\n
The prior distributions for the magnitude of the noise spectral components are modeled as Rayleigh probability density function (PDF) with scale parameter\n\n\nσ\n\n, which is fitted to the observations by a maximum likelihood procedure [16, 17], i.e.,
where the auxiliary variable s is defined as \n\ns\n=\nln\n\n\n\n1\nN\n\n\n∑\n\ni\n=\n1\n\nN\n\n\nx\ni\n\n\n\n−\n\n1\nN\n\n\n∑\n\ni\n=\n1\n\nN\n\nln\n\n\nx\ni\n\n\n\n.
\n
The regularization term for the speech samples is defined as (by applying negative logarithm in both sides of (44))
The proposed multiplicative nonnegative matrix factorization method is summarized in Algorithm 4 [16]. In general, like in the specific case of Algorithm 4, one can only guarantee the monotonous descent of the iteration through a majorization-minimization approach [19] or the convergence to a stationary point [20].
\n
The subscript(s) with parenthesis represents corresponding columns or rows of the matrix assigned to a given source. 1 is an approximately sized matrices of ones, and \n\n⊗\n\n represents element-wise multiplication.
The speech and noise audio samples were taken from NOIZEUS [21]. Sampling frequency is 8 KHz. The algorithm is iterated until convergence [16]. The proposed method was compared with Euclidean NMF (EUC-NMF) [5], Itakura-Saito NMF (IS-NMF) [22], posterior regularization NMF (PR-NMF) [15], Wiener filtering [23], and constrained version of NMF (CNMF)[24]. These methods are implemented by considering nonstationary noise, babble noise and street noise. The performance of proposed method was evaluated by using perceptual evaluation of speech quality (PESQ) [25] and source-to-distortion ratio (SDR) [26]. SDR gives the average quality of separation on dB scale and considers signal distortion as well as noise distortion. For PESQ and SDR, the higher value indicates the better performance. Tables 1 and 2 show the PESQ and SDR values of different NMF algorithms evaluated. The experimental results show that proposed method performs better than other existing methods in terms of the PESQ and SDR indices.
\n
Table 1.
PESQ and SDR for babble noise.
\n
Table 2.
PESQ and SDR for street noise.
\n
\n
\n
6. Conclusion
\n
A novel speech enhancement method based on an iterative and regularized NMF algorithm for single-channel source separation is proposed. The clean speech and noise magnitude spectra are modeled as Gamma and Rayleigh distributions, respectively. The corresponding log-likelihood functions are used as penalties to regularize the cost function of the NMF. The estimation of basis matrices and excitation matrices are calculated by using proposed regularization of multiplicative update rules. The experiments reveal that the proposed speech enhancement method outperforms other existing benchmark methods in terms of SDR and PESQ values.
\n
\n\n',keywords:"nonnegative matrix factorization (NMF), speech enhancement, signal-to-noise ratio (SNR), expectation maximization (EM) algorithms, posterior regularization (PR)",chapterPDFUrl:"https://cdn.intechopen.com/pdfs/67368.pdf",chapterXML:"https://mts.intechopen.com/source/xml/67368.xml",downloadPdfUrl:"/chapter/pdf-download/67368",previewPdfUrl:"/chapter/pdf-preview/67368",totalDownloads:277,totalViews:0,totalCrossrefCites:0,totalDimensionsCites:0,hasAltmetrics:0,dateSubmitted:"October 6th 2018",dateReviewed:"February 5th 2019",datePrePublished:"May 27th 2019",datePublished:"February 26th 2020",dateFinished:"May 27th 2019",readingETA:"0",abstract:"Over the years, miscellaneous methods for speech enhancement have been proposed, such as spectral subtraction (SS) and minimum mean square error (MMSE) estimators. These methods do not require any prior knowledge about the speech and noise signals nor any training stage beforehand, so they are highly flexible and allow implementation in various situations. However, these algorithms usually assume that the noise is stationary and are thus not good at dealing with nonstationary noise types, especially under low signal-to-noise (SNR) conditions. To overcome the drawbacks of the above methods, nonnegative matrix factorization (NMF) is introduced. NMF approach is more robust to nonstationary noise. In this chapter, we are actually interested in the application of speech enhancement using NMF approach. A speech enhancement method based on regularized nonnegative matrix factorization (NMF) for nonstationary Gaussian noise is proposed. The spectral components of speech and noise are modeled as Gamma and Rayleigh, respectively. We propose to adaptively estimate the sufficient statistics of these distributions to obtain a natural regularization of the NMF criterion.",reviewType:"peer-reviewed",bibtexUrl:"/chapter/bibtex/67368",risUrl:"/chapter/ris/67368",book:{slug:"new-frontiers-in-brain-computer-interfaces"},signatures:"Sunnydayal Vanambathina",authors:[{id:"279891",title:"Dr.",name:"Sunnydayal",middleName:null,surname:"Vanambathina",fullName:"Sunnydayal Vanambathina",slug:"sunnydayal-vanambathina",email:"sunny.conference@gmail.com",position:null,institution:null}],sections:[{id:"sec_1",title:"1. Introduction",level:"1"},{id:"sec_2",title:"2. Nonnegative matrix factorization",level:"1"},{id:"sec_2_2",title:"2.1 NMF model",level:"2"},{id:"sec_3_2",title:"2.2 Optimization formulation",level:"2"},{id:"sec_4_2",title:"2.3 Parameter estimation",level:"2"},{id:"sec_6",title:"3. A probabilistic latent variable model with time-frequency constraints",level:"1"},{id:"sec_6_2",title:"3.1 Posterior regularization",level:"2"},{id:"sec_7_2",title:"3.2 Linear grouping expectation constraints",level:"2"},{id:"sec_8_2",title:"3.3 Parameter estimation",level:"2"},{id:"sec_10",title:"4. An iterative posterior NMF method for speech enhancement in the presence of additive Gaussian noise (proposed algorithm)",level:"1"},{id:"sec_10_2",title:"4.1 Notation and basic concepts",level:"2"},{id:"sec_11_2",title:"4.2 Proposed regularization",level:"2"},{id:"sec_13",title:"5. Experimental results",level:"1"},{id:"sec_14",title:"6. Conclusion",level:"1"}],chapterReferences:[{id:"B1",body:'Lee DD, Seung HS. Learning the parts of objects by non-negative matrix factorization. Nature. 1999;401:788-791\n'},{id:"B2",body:'Smaragdis P. Non-negative matrix factorization for polyphonic music transcription. IEEE Workshop on Applications of Signal Processing to Audio and Acoustics; 19–22 October 2003. Mohonk Mountain; 2013. pp. 177-180\n'},{id:"B3",body:'Bryan NJ, Mysore GJ. An efficient posterior regularized latent variable model for interactive sound source separation. In: International Conference on Machine Learning (ICML); June 2013\n'},{id:"B4",body:'Boyd S, Vandenberghe L. Convex Optimization. New York, NY, USA: Cambridge University Press; 2004\n'},{id:"B5",body:'Lee DD, Seung HS. Algorithms for Non-negative Matrix Factorization. NIPS Proceedings. 2001\n'},{id:"B6",body:'Hunter DR, Lange K. A tutorial on MM algorithms. The American Statistician. 2004;58:30-37\n'},{id:"B7",body:'Paltz N. Separation by ‘Humming’: user-guided sound extraction from monophonic mixtures. In: IEEE Workshop on Applications of Signal Processing to Audio and Acoustics (WASPAA); 2009. pp. 69–72\n'},{id:"B8",body:'Fitzgerald D. User assisted separation using tensor factorisations. In: European Signal Processing Conference (EUSIPCO). 2012. pp. 2412–2416\n'},{id:"B9",body:'Graca J, Ganchev K, Taskar B. Expectation maximization and posterior constraints. Advances in Neural Information Processing Systems. 2008;20:1-8\n'},{id:"B10",body:'Ganchev K, Gillenwater J. Posterior regularization for structured latent variable models. Journal of Machine Learning Research. 2010;11:2001-2049\n'},{id:"B11",body:'Graça J, Ganchev K, Taskar B, Pereira F. Posterior vs. parameter sparsity in latent variable models. NIPS–Advances in Neural Information Processing Systems. 2009:664-672\n'},{id:"B12",body:'Smaragdis P, Raj B. Shift-invariant probabilistic latent component analysis. Journal of Machine Learning Research. Technical Report TR2007009, MERL; December, 2007:5\n'},{id:"B13",body:'Mysore GJ, Smaragdis P. A non-negative approach to semi-supervised separation of speech from noise with the use of temporal dynamics Gautham J. Mysore Advanced Technology Labs Adobe Systems Inc, University of Illinois at Urbana-Champaign, Adobe Systems Inc. IEEE International Conference on Acoustics, Speech and Signal Processing–ICASSP 2011; 2011. pp. 17–20\n'},{id:"B14",body:'Bertin N, Badeau R, Vincent E. Enforcing harmonicity and smoothness in Bayesian non-negative matrix factorization applied to polyphonic music transcription. IEEE Transactions on Audio, Speech and Language Processing. 2010;18:538-549\n'},{id:"B15",body:'Bryan NJ, Mysore GJ. An Efficient Posterior Regularized Latent Variable Model for Interactive Sound Source Separation. in Icml, 2013\n'},{id:"B16",body:'Sunnydayal K k, Cruces-Alvarez SA. An iterative posterior NMF method for speech enhancement in the presence of additive Gaussian noise. Neurocomputing. 2017;230:312-315\n'},{id:"B17",body:'Cruces-Alvarez SA, Cichocki A, ichi Amari S. From blind signal extraction to blind instantaneous signal separation: Criteria, algorithms, and stability. IEEE Transactions on Neural Networks. 2004;15:859-873\n'},{id:"B18",body:'Erkelens JS, Hendriks RC, Heusdens R, Jensen J. Minimum mean-square error estimation of discrete Fourier coefficients with generalized gamma priors. IEEE Transactions on Audio, Speech and Language Processing. 2007;15(6):1741-1752\n'},{id:"B19",body:'Cichocki A, Cruces S, ichi Amari S. Generalized alpha-beta divergences and their application to robust nonnegative matrix factorization. Entropy;13:134-170\n'},{id:"B20",body:'Lin C-J. On the convergence of multiplicative update for nonnegative matrix factorization. IEEE Transactions on Neural Networks. 2007;18:1589-1596\n'},{id:"B21",body:'\nhttps://ecs.utdallas.edu/loizou/speech/noizeus/ [Online]\n'},{id:"B22",body:'Févotte C, Bertin N, Durrieu J-L. Nonnegative matrix factorization with the Itakura-Saito divergence: With application to music analysis. Neural Computation. 2009;21:793-830\n'},{id:"B23",body:'Ephraim Y, Malah D. Speech enhancement using a minimum mean-square error short-time spectral amplitude estimator. IEEE Transactions on Acoustics. 1984;32:1109-1121\n'},{id:"B24",body:'Berry MW, Browne M, Langville AN, Pauca VP, Plemmons RJ. Algorithms and applications for approximate nonnegative matrix factorization. Computational Statistics and Data Analysis. 2007;52(1):155-173\n'},{id:"B25",body:'Hu Y, Loizou PC. Evaluation of objective quality measures for speech enhancement. IEEE Transactions on Acoustics, Speech, and Signal Processing. 2008;16(1):229-238\n'},{id:"B26",body:'Vincent E, Gribonval R, Fevotte C. Performance measurement in blind audio source separation. IEEE Transactions on Audio, Speech and Language Processing. 2006;14:1462-1469\n'}],footnotes:[],contributors:[{corresp:"yes",contributorFullName:"Sunnydayal Vanambathina",address:"sunny.conference@gmail.com",affiliation:'
Department of Electronics and Communication Engineering, VIT-AP University, Amaravati, India
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1. Introduction
Smart materials, which are also called intelligent materials, are engineered materials that have the ability to respond to the changes that occur around them in a controlled fashion by external stimuli, such as stress, heat, light, ultraviolet, moisture, chemical compounds, mechanical strength, and electric and magnetic fields. We can simply define smart materials as materials which adapt themselves as per required condition. The history of the discovery of these materials dates back to the 1880s when Jacques and Pierre Curie noticed a phenomenon that pressure generates electrification around a number of minerals such as quartz and tourmaline, and this phenomenon is called piezoelectric effect, so the piezoelectric materials are the oldest type of smart materials, which are utilized extensively in the fabrication of various devices such as transducers, sensors, actuators, surface acoustic wave devices, frequency control, etc. There are a lot of smart material types like piezoelectric materials, thermochromic pigments, shape memory alloys, magnetostrictive, shape memory polymers, hydrogels, electroactive polymers and bi-component fibers, etc.
Anisotropic smart structures (ASSs) are getting great attention of researchers due to their applications in textile, aerospace, mass transit, marine, automotive, computers and other electronic industries, consumer goods applications, mechanical and civil engineering, infertility treatment, micropumps, medical equipment applications, ultrasonic micromotors, microvalves and photovoltaics, rotating machinery applications, and much more [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12].
The classical thermoelasticity (CTE) theory of Duhamel [13] and Neumann [14] has two shortcomings based on parabolic heat conduction equation of this theory: the first does not involve any elastic terms, while the second has infinite propagation speeds of thermoelastic waves. In order to overcome the first shortcoming, Biot [15] proposed the classical coupled thermoelasticity (CCTE). But CTE and CCTE have the second shortcoming. So, several generalized thermoelasticity theories have been developed to overcome the second shortcoming of CTE. Among of these theories are Lord and Shulman (LS) [16], Green and Lindsay (GL) [17], and Green and Naghdi [18, 19] theories of thermoelasticity with and without energy dissipation, dual-phase-lag thermoelasticity (DPLTE) [20, 21] and three-phase-lag thermoelasticity (TPLTE) [22]. Although thermoelastic phenomena in the majority of practical applications are adequately modeled with the classical Fourier heat conduction equation, there are an important number of problems that require consideration of nonlinear heat conduction equation. It is appropriate in these cases to apply the nonlinear generalized theory of thermoelasticity; great attention has been paid to investigate the nonlinear generalized thermoelastic problems by using numerical methods [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34]. Fahmy [35, 36, 37, 38, 39] introduced the mathematical foundations of three-temperature (3T) field to thermoelasticity.
The fractional calculus is the mathematical branch that used to study the theory and applications of derivatives and integrals of arbitrary non-integer order. This branch has emerged in recent years as an effective tool for modeling and simulation of various engineering and industrial applications [40, 41]. Due to the nonlocal nature of fractional order operators, they are useful for describing the memory and hereditary properties of various materials and processes. Also, the fractional calculus has drawn wide attention from the researchers of various countries in recent years due to its applications in solid mechanics, fluid dynamics, viscoelasticity, heat conduction modeling and identification, biology, food engineering, econophysics, biophysics, biochemistry, electrochemistry, electrical engineering, finance and control theory, robotics and control theory, signal and image processing, electronics, electric circuits, wave propagation, nanotechnology, etc. [42, 43, 44].
Several mathematics researchers have contributed to the history of fractional calculus, where Euler mentioned interpolating between integral orders of a derivative in 1730. Then, Laplace defined a fractional derivative by means of an integral in 1812.
Lacroix presented the first formula for the fractional order derivative appeared in 1819, where he introduced the nth derivative of the function y=xm as follows:
dndxn=Γm+1Γm‐n+1xm‐nE1
Liouville supposed that dvdxveax=aveaxforv>0 to get the following fractional order derivative:
dvx‐adxv=−1vΓa+vΓax‐a‐vE2
By using Cauchy’s integral formula for complex valued analytical functions, Laurent defined the integration of arbitrary order v>0 as
cDxvfx=Dcxm‐ρfx=dmdxm1Γρ∫cxx‐tρ‐1ftdt,0<ρ≤1E3
where cDxv denotes differentiation of order v of the function f along the x-axis.
Cauchy presented the following fractional order derivative:
f+α=∫fτt−τ−α−1Γ−adτE4
In 1967, the Italian mathematician Caputo presented his fractional derivative of order α>0 as
D∗αft=1Γm−α∫0tfmτt−τα+1−mdτ,m−1<α<m,α<0E5
Diethelm [45] has suggested the Caputo derivative to be in the following form:
Daζfτ=∫aτKζτ‐ξfmξdξE6
where f(m) is the mth order derivative and m is an integer such that m‐1<ζ≤m
Kζτ‐ξ=τ‐ξm‐ζ‐1Γm‐ζE7
Wang and Li [46] have introduced a memory-dependent derivative (MDD)
Dωζfτ=1ω∫τ‐ωτKζτ‐ξfmξdξE8
where the first-order ζ=1 of MDD for a differentiable function fτ can be expressed as
Dωfτ=1ω∫τ‐ωτKτ‐ξf′ξdξE9
Based on several practical applications, the memory effect needs weight 0≤Kτ‐ξ<1 for ξ∈τ‐ωτ, so the MDD magnitude Dωfτ is usually smaller than f′τ, where the time delay ω>0 and the kernel function (0≤Kτ‐ξ≤1 for ξ∈τ‐ξτ) can be chosen arbitrarily on the delayed interval τ‐ωτ, the practical kernel functions are 1,1−τ‐ξ and 1−τ‐ξωp, p=14,1,2, etc. These functions are monotonically increasing with K=0 for the past time τ‐ξ and K=1 for the present time τ. The main feature of MDD is that the real-time functional value depends also on the past time [τ‐ξ‐τ[. So, Dω depends on the past time (nonlocal operator), while the integration does not depend on the past time (local operator).
As a special case Kτ‐ξ≡1, we have
Dωfτ=1ω∫τ−ωτf′ξdξ=fτ−fτ−ωω→f′τE10
The above equation shows that the common derivative ddτ is the limit of Dω as ω→0. That is,
Dωfτ≤∂f∂τ=limω→0fτ+ω−fτωE11
Now, the boundary element method (BEM) [47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80] is widely adopted for solving several engineering problems due to its easy implementation. In the BEM, only the boundary of the domain needs to be discretized, so it has a major advantage over other methods requiring full domain discretization [81, 82, 83, 84, 85, 86, 87] such as finite difference method (FDM), finite element method (FEM), and finite volume method (FVM) in engineering applications. This advantage of BEM over domain methods has significant importance for modeling of nonlinear generalized thermoelastic problems which can be implemented using BEM with little cost and less input data. Previously scientists have proven that FEM covers more engineering applications than BEM which is more efficient for infinite domain problems. But currently BEM scientists have changed their thinking and vision on BEM, where the BEM researchers developed the BEM technique for solving inhomogeneous and nonlinear problems involving infinite and semi-infinite domains by using a lot of software like FastBEM and ExaFMM.
The main objective of this chapter is to introduce a novel memory-dependent derivative model for solving transient three-temperature nonlinear thermal stress problems in functionally graded anisotropic (FGA) smart structures. The governing equations of the considered model are nonlinear and very difficult if not impossible to solve analytically. Therefore, we develop a new efficient boundary element technique for solving such equations. Numerical results show the effects of MDD on the three-temperature distributions and the influence of MDD and anisotropy on the nonlinear thermal stresses of FGA smart structures. Also, numerical results demonstrate the validity and accuracy of the proposed model.
A brief summary of the chapter is as follows: Section 1 introduces the background and provides the readers with the necessary information to books and articles for a better understanding of smart material problems, memory-dependent derivative history, and their applications. Section 2 describes the physical modeling of memory-dependent derivative problems of three-temperature nonlinear thermal stresses in FGA structures. Section 3 outlines the BEM implementation for obtaining the temperature field of the considered problem. Section 4 outlines the BEM implementation for obtaining the dispacement field of the considered problem. Section 5 introduces computing performance of the proposed model. Section 6 presents the new numerical results that describe the effects of memory-dependent derivative and anisotropy on the problem’s field variations. Lastly, Section 7 outlines the significant findings of this chapter.
2. Formulation of the problem
With reference to a Cartesian system x1x2x3 with a configuration R bounded by a closed surface S as shown in Figure 1.
Figure 1.
Computational domain of the considered smart structure.
The governing equations for the transient three-temperature nonlinear thermal stresses problems of FGA smart structures with memory-dependent derivatives can be written as [35].
σij,j+ρFi=πüiE12
Di,i=0E13
where
σij=x+1mCijkleδij‐βabTα‐Tα0+τ1ṪαE14
Di=x+1meijkεjk+fikEkεij=12ui,j+uj,i,E15
where σij, Fi, εij, εijk, ui, and ρ are the force stress tensor, mass force vector, strain tensor, alternate tensor, displacement vector, and density, respectively, CijklCijkl=Cklij=Cjikl is the constant elastic moduli, e is the dilatation, βijβij=βji are the stress-temperature coefficients, Di is the electric displacement, eijk is the piezoelectric tensor, fik is the permittivity tensor, and Ek is the electric field vector.
The two-dimensional three-temperature (2D-3T) radiative heat conduction equations can be expressed as
ce∂Terτ∂τ−1ρ∇Ke∇Terτ=−WeiTe−Ti−WepTe−TpE16
ci∂Tirτ∂τ−1ρ∇Ki∇Tirτ=WeiTe−TiE17
cpTp3∂Tprτ∂τ−1ρ∇Kp∇Tprτ=WepTe−TpE18
where e,i,∧p denote electron, ion, and phonon, respectively; cecicp, KeKiKp, and TeTiTp are specific heat capacities, conductive coefficients, and temperature functions, respectively; Wei is the electron-ion coefficient; and Wep is the electron-phonon coefficient.
3. BEM solution of temperature field
This section concerns using a boundary element method to solve the temperature model.
The above 2D-3T radiative heat conduction Eqs. (16)-(18) can be expressed in the context of nonlinear thermal stresses of FGA smart structures as in [36].
The global matrix equation for all i nodes can be expressed as
HQ=GP+DΘ+FSE77
where Q is the displacement vector, P is the traction vector, Θ is the electric potential vector, and S is the electric potential gradient vector.
Substituting the boundary conditions into (77), we obtain the following system of equations:
AX=BE78
We apply an explicit staggered algorithm to solve the system (78) and obtain the temperature and displacement fields as follows:
From Eq. (48) we obtain the temperature field in terms of the displacement field.
We predict the displacement field and solve the resulted equation for the temperature field.
We correct the displacement field using the computed temperature field for Eq. (78).
An explicit staggered algorithm based on communication-avoiding Arnoldi as described in Hoemmen [91] is very suitable for efficient implementation in Matlab (R2019a) with the aim of specifically improving its performance for the solution of the resulting linear algebraic systems.
5. Computational performance of the problem
According to Fahmy [35], the computer performance with simulation can be computed based on account and communication process, elements underlying the hardware and functional computation. The main objective of our proposed technique during simulation process is to use the preconditioners which are efficient to improve the overall CPU utilization of the cluster, accelerate the iterative method, and reduce the input/output and the interprocessor communication costs. Also, Fahmy [35] compared the communication-avoiding Krylov methods that are based on the s-step Krylov methods such as communication-avoiding generalized minimal residual (CA-GMRES) of Saad and Schultz [92], communication-avoiding Arnoldi (CA-Arnoldi) of the Arnoldi [93] and communication-avoiding Lanczos (CA-Lanczos) of Lanczos [94], with their corresponding standard Krylov methods. CA-Arnoldi which is also called Arnoldi (s, t) algorithm is different from standard Arnoldi (s) st=1, where s is the number of inner iteration steps and t is the number of outer iteration steps. According to [35], the CA-Arnoldi has numerical stability, convergence, and performance due to the implementation of algorithm shown in Figure 2, which is based on the QR factorization update and block classical Gram-Schmidt (block CGS) approach or block modified Gram-Schmidt (block MGS) approach where
Figure 2.
CA-Arnoldi iteration algorithm.
Vk=vsk+1vsk+2…vsk+sE79
and
Qk=Q0Q1…Qk−1E80
The generalized minimal residual (GMRES) method of Saad and Schultz [92] is a Krylov subspace method for solving nonsymmetric linear systems. The CA-GMRES algorithm is based on Arnoldi (s, t) and equivalent to standard GMRES in exact arithmetic. Also, the GMRES or CA-GMRES are convergent at the same rate for problems, but Hoemmen [91] proved that CA-GMRES algorithm shown in Figure 3 converges for the s-step basis lengths and restart lengths used for obtaining maximum performance. Lanczos method can be considered as a special case of Arnoldi method for symmetric and real case of A or Hermitian and complex case of A. Symmetric Lanczos which is also called Lanczos is different from nonsymmetric Lanczos. We implemented a communication-avoiding version of symmetric Lanczos (CA-Lanczos) for solving symmetric positive definite (SPD) eigenvalue problems. Also, we implement CA-Lanczos iteration algorithm shown in Figure 4, which is also called Lanczos (s, t), where s is the s-step basis length and t is the outer iterations number before restart. This algorithm is based on using rank revealing-tall skinny QR-block Gram-Schmidt (RR-TSQR-BGS) orthogonalization method which connects between TSQR and block Gram-Schmidt, where we have been using the right-shifted basis matrix at outer iteration k as follows:
Figure 3.
CA-GMRES iteration algorithm.
Figure 4.
CA-Lanczos iteration algorithm.
Vk′=Vsk+2…vsk+sE81
and
Vk′=Vk′vsk+s+1E82
For more details about the considered preconditioners and algorithms, we refer the interested readers to [91].
The main objective of this section is to implement an accurate and robust preconditioning technique for solving the dense nonsymmetric algebraic system of linear equations arising from the BEM. So, a communication-avoiding Arnoldi of the Arnoldi [93] has been implemented for solving the resulting linear systems in order to reduce the iteration number and CPU time. The BEM discretization is employed in 1280 quadrilateral elements, with 3964 degrees of freedom (DOF). A comparative performance of preconditioned Krylov subspace solvers (CA-Arnoldi, CA-GMRES, and CA-Lanczos) has been shown in Table 1, where the number of DOF is 3964 and “–” was defined as the divergence process. From the results of Table 1. The CA-Arnoldi, CA-GMRES, and CA-Lanczos are more cost-effective than the other Krylov subspace methods Arnoldi, GMRES, and Lanczos, respectively. Also, CA-Arnoldi, CA-GMRES, and CA-Lanczos have been compared with each other in Table 2. It can be seen from this table that the performance of CA-Arnoldi is superior than the other iterative methods.
Methods
Preconditioning techniques
Iterations
Residual
Time of each iterative step (s)
Time of solution
Direct methods
NO
—
—
—
9 min 50 s
Arnoldi
NO
174
7.21E–07
3.85
11 min 25 s
JOBI
26
5.22E–07
3.86
2 min 38 s
BJOB
22
1.34E–06
3.86
2 min 23 s
ILU3
47
1.66E–06
3.84
4 min 2 s
ILU5
48
1.38E–06
3.89
4 min 6 s
DILU
48
1.53E–06
5.45
4 min 18 s
CA–Arnoldi
NO
360
6.96E–07
1.95
11 min 53 s
JOBI
20
4.42E–07
1.96
1 min 30 s
BJOB
20
2.30E–08
1.96
1 min 30 s
ILU3
40
7.87E–07
1.96
2 min 11 s
ILU5
60
1.28E–08
1.96
2 min 48 s
DILU
60
1.59E–07
3.07
4 min 1 s
GMRES
NO
280
2.36E–08
1.90
6 min 20 s
JOBI
40
5.01E–13
1.91
2 min 10 s
BJOB
40
2.05E–11
1.91
2 min 10 s
ILU3
40
4.70E–08
1.91
2 min 10 s
ILU5
40
3.13E–08
2.60
2 min 10 s
DILU
40
6.19E–08
3.07
2 min 48 s
CA–GMRES
NO
120
6.89E–07
3.78
7 min 57 s
JOBI
12
1.00E–05
3.76
1 min 41 s
BJOB
12
2.22E–06
3.76
1 min 42 s
ILU3
26
3.63E–06
3.75
2 min 34 s
ILU5
22
4.05E–06
3.75
2 min 20 s
DILU
25
5.19E–06
5.93
3 min 18 s
Lanczos
NO
135
7.24E–07
3.80
8 min 41 s
JOBI
22
4.87E–07
3.75
2 min 33 s
BJOB
18
9.27E–07
5.18
3 min 2 s
ILU3
42
2.41E–07
3.81
3 min 48 s
ILU5
36
6.41E–07
3.78
3 min 18 s
DILU
38
2.04E–07
5.00
3 min 32 s
CA–Lanczos
NO
129
1.30E–04
3.75
9 min 22 s
JOBI
16
8.64E–07
3.76
2 min 3s
BJOB
14
1.69E–07
3.77
2 min 0 s
ILU3
24
9.29E–07
3.87
2 min 31 s
ILU5
31
1.91E–07
3.90
3 min 1 s
DILU
27
8.11E–07
5.95
3 min 31 s
Table 1.
Performances of preconditioned Krylov subspace iterative methods for DOF 3964.
Solvers
DOF
965
1505
3380
3964
6005
CA–Arnoldi
Residual
6.81E–12
5.38E–12
4.13E–11
4.17E–11
7.57E–11
CPU time (s)
4.96
10.78
99.24
134.26
293.29
Iterations
25
25
25
25
25
CA–GMRES
Residual
2.98E–12
1.90E–12
1.28E–11
1.36E–11
1.22E–11
CPU time (s)
5.06
11.49
126.38
164.09
445.51
Iterations
50
50
50
50
50
CA– Lanczos
Residual
7.20E–11
3.35E–11
2.72E–11
3.97E–11
8.33E–11
CPU time (s)
5.05
11.47
139.07
180.49
514.72
Iterations
22
26
28
30
32
Table 2.
The CPU time and the number of iterations for some communication–avoiding Krylov subspace solvers.
6. Numerical results and discussion
In order to illustrate the numerical results of this study, we consider a monoclinic graphite-epoxy as an anisotropic smart material which has the following constants [35].
Mass density ρ=7820kg/m2 and heat capacity c=461J/kgk.
The technique that has been proposed in the current chapter can be applicable to a wide range of three-temperature nonlinear thermal stress problems of FGA structures. The main aim of this chapter is to assess the impact of MDD and anisotropy on the three-temperature nonlinear thermal stress distributions.
The proposed technique that has been implemented in the current study can be applicable to a wide variety of FGA smart structure problems involving three temperatures. All the physical parameters satisfy the initial and boundary conditions. The efficiency of our BEM modeling technique has been improved using an explicit staggered algorithm based on communication-avoiding Arnoldi procedure to decrease the computation time.
Figure 5 shows the variations of the three temperatures Te,TiandTp with the time τ in the presence of MDD. Figure 6 shows the variations of the three temperatures Te,TiandTp with the time τ in the presence of MDD. It can be seen from Figures 5 and 6 that the MDD has a significant effect on the temperature distributions.
Figure 5.
Variation of the three-temperature (with memory) with time τ.
Figure 6.
Variation of the three-temperature (without memory) with time τ.
In order to study the anisotropy and MDD effects on the nonlinear thermal stresses, we assume the following four cases: A, B, C, and D, where case A denotes the nonlinear thermal stress distribution in the isotropic material without MDD effect, case B denotes the nonlinear thermal stress distribution in isotropic material with MDD effect, case C denotes the nonlinear thermal stress distribution in anisotropic material without MDD effect, and case D denotes nonlinear thermal stress distribution in anisotropic material with MDD effect.
Figures 7–9 show the variation of the nonlinear thermal stresses σ11,σ12andσ22 with the time τ. It is clear from these figures that both anisotropy and MDD have a significant influence on the nonlinear thermal stress distributions.
Figure 7.
Variation of the nonlinear thermal stress σ11 with time τ.
Figure 8.
Variation of the nonlinear thermal stress σ12 with time τ.
Figure 9.
Variation of the nonlinear thermal stress σ22 with time τ.
Since there are no available results for the considered problem in the literature. Therefore, we only considered the one-dimensional special case for the variations of the nonlinear thermal stress σ11 with the time τ as shown in Figure 10. The validity and accuracy of our proposed technique was confirmed by comparing graphically our BEM results with those obtained using the FDM of Pazera and Jędrysiak [95] and FEM of Xiong and Tian [96] results based on replacing one-temperature heat conduction with the total three-temperature TT=Te+Ti+Tr heat conduction. It can be noticed that the BEM results are found to agree very well with the FDM and FEM results.
Figure 10.
Variation of the nonlinear thermal stress σ11 with time τ.
7. Conclusion
The main aim of this chapter is to introduce a new MDD model based on BEM for obtaining the transient three-temperature nonlinear thermal stresses in FGA smart structures. The governing equations of this model are very hard to solve analytically because of nonlinearity and anisotropy. To overcome this, we propose a new boundary element formulation for solving such equations. Since the CA kernels of the s-step Krylov methods are faster than the kernels of standard Krylov methods. Therefore, we used an explicit staggered algorithm based on CA-Arnoldi procedure to solve the resulted linear equations. The computational performance of the proposed technique has been performed using communication-avoiding Arnoldi procedure. The numerical results are presented highlighting the effects of MDD on the three-temperature distributions and the influence of MDD and anisotropy on the nonlinear thermal stresses of FGA smart structures. The numerical results also demonstrate the validity and accuracy of the proposed technique. It can be concluded from numerical results of our current general problem that all generalized and nonlinear generalized thermoelasticity theories can be combined with the three-temperature radiative heat conduction to describe the deformation of FGA smart structures in the context of memory-dependent derivatives. From the research that has been performed, it is possible to conclude that the proposed BEM technique is effective and stable for transient three-temperature thermal stress problems in FGA smart structures.
The numerical results for our complex and general problem can provide data references for computer scientists and engineers, geotechnical and geothermal engineers, designers of new materials, and researchers in material science as well as for those working on the development of anisotropic smart structures. In the application of three-temperature theories in advanced manufacturing technologies, with the development of soft machines and robotics in biomedical engineering and advanced manufacturing, transient thermal stresses will be encountered more often where three-temperature radiative heat conduction will turn out to be the best choice for thermomechanical analysis in the design and analysis of advanced smart materials and structures.
\n',keywords:"boundary element method, memory-dependent derivative, three-temperature, nonlinear thermal stresses, FGA smart structures",chapterPDFUrl:"https://cdn.intechopen.com/pdfs/72846.pdf",chapterXML:"https://mts.intechopen.com/source/xml/72846.xml",downloadPdfUrl:"/chapter/pdf-download/72846",previewPdfUrl:"/chapter/pdf-preview/72846",totalDownloads:95,totalViews:0,totalCrossrefCites:0,dateSubmitted:"April 13th 2020",dateReviewed:"May 14th 2020",datePrePublished:"November 28th 2020",datePublished:"November 26th 2020",dateFinished:"July 17th 2020",readingETA:"0",abstract:"The main objective of this chapter is to introduce a novel memory-dependent derivative (MDD) model based on the boundary element method (BEM) for solving transient three-temperature (3T) nonlinear thermal stress problems in functionally graded anisotropic (FGA) smart structures. The governing equations of the considered study are nonlinear and very difficult if not impossible to solve analytically. Therefore, we develop a new boundary element scheme for solving such equations. The numerical results are presented highlighting the effects of the MDD on the temperatures and nonlinear thermal stress distributions and also the effect of anisotropy on the nonlinear thermal stress distributions in FGA smart structures. The numerical results also verify the validity and accuracy of the proposed methodology. The computing performance of the proposed model has been performed using communication-avoiding Arnoldi procedure. We can conclude that the results of this chapter contribute to increase our understanding on the FGA smart structures. Consequently, the results also contribute to the further development of technological and industrial applications of FGA smart structures of various characteristics.",reviewType:"peer-reviewed",bibtexUrl:"/chapter/bibtex/72846",risUrl:"/chapter/ris/72846",signatures:"Mohamed Abdelsabour Fahmy",book:{id:"10049",title:"Advanced Functional Materials",subtitle:null,fullTitle:"Advanced Functional Materials",slug:"advanced-functional-materials",publishedDate:"November 26th 2020",bookSignature:"Nevin Tasaltin, Paul Sunday Nnamchi and Safaa Saud",coverURL:"https://cdn.intechopen.com/books/images_new/10049.jpg",licenceType:"CC BY 3.0",editedByType:"Edited by",editors:[{id:"94825",title:"Associate Prof.",name:"Nevin",middleName:null,surname:"Tasaltin",slug:"nevin-tasaltin",fullName:"Nevin Tasaltin"}],productType:{id:"1",title:"Edited Volume",chapterContentType:"chapter",authoredCaption:"Edited by"}},authors:[{id:"233766",title:"Prof.",name:"Mohamed Abdelsabour",middleName:"Abdelsabour",surname:"Fahmy",fullName:"Mohamed Abdelsabour Fahmy",slug:"mohamed-abdelsabour-fahmy",email:"mohamed_fahmy@ci.suez.edu.eg",position:null,institution:{name:"Suez Canal University",institutionURL:null,country:{name:"Egypt"}}}],sections:[{id:"sec_1",title:"1. Introduction",level:"1"},{id:"sec_2",title:"2. Formulation of the problem",level:"1"},{id:"sec_3",title:"3. BEM solution of temperature field",level:"1"},{id:"sec_4",title:"4. BEM solution of displacement field",level:"1"},{id:"sec_5",title:"5. Computational performance of the problem",level:"1"},{id:"sec_6",title:"6. Numerical results and discussion",level:"1"},{id:"sec_7",title:"7. Conclusion",level:"1"}],chapterReferences:[{id:"B1",body:'Fahmy MA. A time-stepping DRBEM for 3D anisotropic functionally graded piezoelectric structures under the influence of gravitational waves. In: Proceedings of the 1st GeoMEast International Congress and Exhibition (GeoMEast 2017); 15–19 July 2017; Sharm El Sheikh, Egypt. Facing the Challenges in Structural Engineering, Sustainable Civil Infrastructures. 2017. pp. 350-365'},{id:"B2",body:'Fahmy MA. 3D DRBEM modeling for rotating initially stressed anisotropic functionally graded piezoelectric plates. In: Proceedings of the 7th European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS 2016); 5–10 June 2016; Crete Island, Greece. pp. 7640-7658'},{id:"B3",body:'Fahmy MA. Boundary element solution of 2D coupled problem in anisotropic piezoelectric FGM plates. In: Proceedings of the 6th International Conference on Computational Methods for Coupled Problems in Science and Engineering (Coupled Problems 2015); 18–20 May 2015; Venice, Italy. 2015. pp. 382-391'},{id:"B4",body:'Fahmy MA. The DRBEM solution of the generalized magneto-thermo-viscoelastic problems in 3D anisotropic functionally graded solids. In: Proceedings of the 5th International Conference on Coupled Problems in Science and Engineering (Coupled Problems 2013); 17–19 June 2013; Ibiza, Spain. 2013. pp. 862-872'},{id:"B5",body:'Fahmy MA. A computerized boundary element model for simulation and optimization of fractional-order three temperatures nonlinear generalized piezothermoelastic problems based on genetic algorithm. In: AIP Conference Proceedings 2138 of Innovation and Analytics Conference and Exhibition (IACE 2019); 25–28 March 2019; Sintok, Malaysia. 2019. p. 030015'},{id:"B6",body:'Huang R, Zheng SJ, Liu ZS, Ng TY. Recent advances of the constitutive models of smart materials—Hydrogels and shape memory polymers. International Journal of Applied Mechanics. 2020;12:2050014'},{id:"B7",body:'Fahmy MA. Shape design sensitivity and optimization of anisotropic functionally graded smart structures using bicubic B-splines DRBEM. Engineering Analysis with Boundary Elements. 2018;87:27-35'},{id:"B8",body:'Sigmund O, Torquato S. Design of smart composite materials using topology optimization. Smart Materials and Structures. 1999;8:365-379'},{id:"B9",body:'Jin B, Zhong Z. A moving mode-III crack in functionally graded piezoelectric material: Permeable problem. Mechanics Research Communications. 2002;29:217-224'},{id:"B10",body:'Lin S, Narita F, Shindo Y. Electroelastic analysis of a penny-shaped crack in a piezo-electric ceramic under mode I loading. Mechanics Research Communications. 2003;30:371-386'},{id:"B11",body:'Liu W, Ma S, Wu H. Three-dimensional analysis of functionally graded piezoelectric plate with arbitrarily distributed material properties. Journal of Wuhan University of Technology. 2014;29(August):712-720'},{id:"B12",body:'Stanak P. Three-dimensional meshless modelling of functionally graded piezoelectric sensor. J Mech Phys Solids Mech. 2014;2013:425-432'},{id:"B13",body:'Duhamel J. Some memoire sur les phenomenes thermo-mechanique. Journal de l’École Polytechnique. 1837;15:1-57'},{id:"B14",body:'Neumann F. Vorlesungen Uber die theorie der elasticitat. Brestau: Meyer; 1885'},{id:"B15",body:'Biot M. Thermoelasticity and irreversible thermo-dynamics. Journal of Applied Physics. 1956;27:249-253'},{id:"B16",body:'Lord HW, Shulman Y. A generalized dynamical theory of thermoelasticity. Journal of the Mechanics and Physics of Solids. 1967;15:299-309'},{id:"B17",body:'Green AE, Lindsay KA. Thermoelasticity. Journal of Elasticity. 1972;2:1-7'},{id:"B18",body:'Green AE, Naghdi PM. On undamped heat waves in an elastic solid. Journal of Thermal Stresses. 1992;15:253-264'},{id:"B19",body:'Green AE, Naghdi PM. Thermoelasticity without energy dissipation. Journal of Elasticity. 1993;31:189-208'},{id:"B20",body:'Tzou DY. A unified field approach for heat conduction from macro to micro scales. ASME Journal of Heat Transfer. 1995;117:8-16'},{id:"B21",body:'Chandrasekharaiah DS. Hyperbolic thermoelasticity: A review of recent literature. Applied Mechanics Reviews. 1998;51:705-729'},{id:"B22",body:'Roychoudhuri SK. On a thermoelastic three-phase-lag model. Journal of Thermal Stresses. 2007;30:231-238'},{id:"B23",body:'Fahmy MA. A time-stepping DRBEM for magneto-thermo-viscoelastic interactions in a rotating nonhomogeneous anisotropic solid. International Journal of Applied Mechanics. 2011;3:1-24'},{id:"B24",body:'Fahmy MA. A time-stepping DRBEM for the transient magneto-thermo-visco-elastic stresses in a rotating non-homogeneous anisotropic solid. Engineering Analysis with Boundary Elements. 2012;36:335-345'},{id:"B25",body:'Fahmy MA. Numerical modeling of transient magneto-thermo-viscoelastic waves in a rotating nonhomogeneous anisotropic solid under initial stress. International Journal of Modeling. Simulation and Scientific Computing. 2012;3:1250002'},{id:"B26",body:'Fahmy MA. Transient magneto-thermo-viscoelastic stresses in a rotating nonhomogeneous anisotropic solid with and without a moving heat source. Journal of Engineering Physics and Thermophysics. 2012;85:950-958'},{id:"B27",body:'Fahmy MA. Transient magneto-thermo-elastic stresses in an anisotropic viscoelastic solid with and without moving heat source. Numerical Heat Transfer, Part A: Applications. 2012;61:547-564'},{id:"B28",body:'Fahmy MA. Transient magneto-thermoviscoelastic plane waves in a non-homogeneous anisotropic thick strip subjected to a moving heat source. Applied Mathematical Modelling. 2012;36:4565-4578'},{id:"B29",body:'Fahmy MA. The effect of rotation and inhomogeneity on the transient magneto-thermoviscoelastic stresses in an anisotropic solid. ASME Journal of Applied Mechanics. 2012;79:1015'},{id:"B30",body:'Fahmy MA. Finite difference algorithm for transient magneto-thermo-elastic stresses in a non-homogeneous solid cylinder. International Journal of Materials Engineering and Technology. 2010;3:87-93'},{id:"B31",body:'Fahmy MA. Boundary element algorithm for nonlinear modeling and simulation of three temperature anisotropic generalized micropolar piezothermoelasticity with memory-dependent derivative. International Journal of Applied Mechanics. 2020;12:2050027'},{id:"B32",body:'Fahmy MA. Thermal stresses in a spherical shell under three thermoelastic models using FDM. International Journal of Numerical methods and Applications. 2009;2:123-128'},{id:"B33",body:'Ezzat MA, El-Karamany AS, Alaa A, El-Bary AA. On dual-phase-lag thermoelasticity theory with memory-dependent derivative. Mechanics of Advanced Materials and Structures. 2017;24:908-916'},{id:"B34",body:'Ezzat MA, El-Karamany AS, Alaa A, El-Bary AA. Generalized thermoelasticity with memory-dependent derivatives involving two temperatures. Mechanics of Advanced Materials and Structures. 2016;23:545-553'},{id:"B35",body:'Fahmy MA. A new boundary element strategy for modeling and simulation of three temperatures nonlinear generalized micropolar-magneto-thermoelastic wave propagation problems in FGA structures. Engineering Analysis with Boundary Elements. 2019;108:192-200'},{id:"B36",body:'Fahmy MA. A new computerized boundary element model for three-temperature nonlinear generalized thermoelastic stresses in anisotropic circular cylindrical plate structures. In: Awrejcewicz J, Grzelczyk D, editors. Dynamical Systems Theory. London, UK: IntechOpen; 2019. pp. 1-17'},{id:"B37",body:'Fahmy MA. Boundary element model for nonlinear fractional-order heat transfer in magneto-thermoelastic FGA structures involving three temperatures. In: Ebrahimi F, editor. Mechanics of Functionally Graded Materials and Structures. IntechOpen: London, UK; 2019. pp. 1-22'},{id:"B38",body:'Fahmy MA. Boundary element mathematical modelling and boundary element numerical techniques for optimization of micropolar thermoviscoelastic problems in solid deformable bodies. In: Sivasankaran S, Nayak PK, Günay E, editors. Mechanics of Solid Deformable Bodies. IntechOpen: London, UK; 2020. pp. 1-21'},{id:"B39",body:'Fahmy MA. Boundary element modeling and optimization based on fractional-order derivative for nonlinear generalized photo-thermoelastic stress wave propagation in three-temperature anisotropic semiconductor structures. In: Sadollah A, Sinha TS, editors. Recent Trends in Computational Intelligence. IntechOpen: London, UK; 2020. pp. 1-16'},{id:"B40",body:'Cattaneo C. Sur une forme de i’equation de la chaleur elinant le paradox d’une propagation instantanc. Comptes Rendus de l\'Académie des Sciences. 1958;247:431-433'},{id:"B41",body:'Oldham KB, Spanier J. The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order. Mineola: Dover Publication; 2006'},{id:"B42",body:'Soukkou A, Belhour MC, Leulmi S. Review, design, optimization and stability analysis of fractional-order PID controller. International Journal of Intelligent Systems Technologies and Applications. 2016;8:73-96'},{id:"B43",body:'Kilbas AA, Srivastava HM, Trujillo JJ. Theory and applications of fractional differential equations. In: Vol. 204 of North-Holland Mathematics Studies. Amsterdam, The Netherlands: Elsevier Science; 2006'},{id:"B44",body:'Sabatier J, Agrawal OP, Machado JAT, editors. Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering. Dordrecht, The Netherlands: Springer; 2007'},{id:"B45",body:'Diethelm K. Generalized compound quadrature formulae for finite-part integrals. IMA Journal of Numerical Analysis. 1997;17:479-493'},{id:"B46",body:'Wang JL, Li HF. Surpassing the fractional derivative: Concept of the memory-dependent derivative. Computers and Mathematics with Applications. 2011;62:1562-1567'},{id:"B47",body:'Fahmy MA. A three-dimensional generalized magneto-thermo-viscoelastic problem of a rotating functionally graded anisotropic solids with and without energy dissipation. Numerical Heat Transfer, Part A: Applications. 2013;63:713-733'},{id:"B48",body:'Fahmy MA. A 2-D DRBEM for generalized magneto-thermo-viscoelastic transient response of rotating functionally graded anisotropic thick strip. International Journal of Engineering and Technology Innovation. 2013;3:70-85'},{id:"B49",body:'Fahmy MA, Salem AM, Metwally MS, Rashid MM. Computer implementation of the DRBEM for studying the generalized thermoelastic responses of functionally graded anisotropic rotating plates with one relaxation time. International Journal of Applied Science and Technology. 2013;3:130-140'},{id:"B50",body:'Fahmy MA, Salem AM, Metwally MS, Rashid MM. Computer implementation of the DRBEM for studying the classical uncoupled theory of thermoelasticity of functionally graded anisotropic rotating plates. International Journal of Engineering Research and Applications. 2013;3:1146-1154'},{id:"B51",body:'Fahmy MA. A computerized DRBEM model for generalized magneto-thermo-visco-elastic stress waves in functionally graded anisotropic thin film/substrate structures. Latin American Journal of Solids and Structures. 2014;11:386-409'},{id:"B52",body:'Fahmy MA, Salem AM, Metwally MS, Rashid MM. Computer implementation of the DRBEM for studying the classical coupled thermoelastic responses of functionally graded anisotropic plates. Physical Science International Journal. 2014;4:674-685'},{id:"B53",body:'Fahmy MA, Salem AM, Metwally MS, Rashid MM. Computer implementation of the DRBEM for studying the generalized thermo elastic responses of functionally graded anisotropic rotating plates with two relaxation times. British Journal of Mathematics & Computer Science. 2014;4:1010-1026'},{id:"B54",body:'Fahmy MA. Computerized Boundary Element Solutions for Thermoelastic Problems: Applications to Functionally Graded Anisotropic Structures. Saarbrücken: LAP Lambert Academic Publishing; 2017'},{id:"B55",body:'Fahmy MA. Boundary Element Computation of Shape Sensitivity and Optimization: Applications to Functionally Graded Anisotropic Structures. Saarbrücken: LAP Lambert Academic Publishing; 2017'},{id:"B56",body:'Fahmy MA. A new computerized boundary element algorithm for cancer modeling of cardiac anisotropy on the ECG simulation. Asian Journal of Research in Computer Science. 2018;2:1-10'},{id:"B57",body:'Brebbia CA, Telles JCF, Wrobel L. Boundary Element Techniques in Engineering. New York: Springer-Verlag; 1984'},{id:"B58",body:'Wrobel LC, Brebbia CA. The dual reciprocity boundary element formulation for nonlinear diffusion problems. Computer Methods in Applied Mechanics and Engineering. 1987;65:147-164'},{id:"B59",body:'Partridge PW, Brebbia CA. Computer implementation of the BEM dual reciprocity method for the solution of general field equations. Communications in Applied Numerical Methods. 1990;6:83-92'},{id:"B60",body:'Partridge PW, Brebbia CA, Wrobel LC. The Dual Reciprocity Boundary Element Method. Southampton: Computational Mechanics Publications; 1992'},{id:"B61",body:'Gaul L, Kögl M, Wagner M. Boundary Element Methods for Engineers and Scientists. Berlin: Springer-Verlag; 2003'},{id:"B62",body:'Zirakashvili N. Solution of contact problems for half-space by boundary element methods based on singular solutions of flamant and boussinesq’s problems. International Journal of Applied Mechanics. 2020;12:2050015'},{id:"B63",body:'Fahmy MA. Implicit-explicit time integration DRBEM for generalized magneto-thermoelasticity problems of rotating anisotropic viscoelastic functionally graded solids. Engineering Analysis with Boundary Elements. 2013;37:107-115'},{id:"B64",body:'Fahmy MA. Generalized magneto-thermo-viscoelastic problems of rotating functionally graded anisotropic plates by the dual reciprocity boundary element method. Journal of Thermal Stresses. 2013;36:1-20'},{id:"B65",body:'Fahmy MA. A 2D time domain DRBEM computer model for magneto-thermoelastic coupled wave propagation problems. International Journal of Engineering and Technology Innovation. 2014;4:138-151'},{id:"B66",body:'Fahmy MA, Al-Harbi SM, Al-Harbi BH. Implicit time-stepping DRBEM for design sensitivity analysis of magneto-thermo-elastic FGA structure under initial stress. American Journal of Mathematical and Computational Sciences. 2017;2:55-62'},{id:"B67",body:'Fahmy MA. The effect of anisotropy on the structure optimization using golden-section search algorithm based on BEM. Journal of Advances in Mathematics and Computer Science. 2017;25:1-18'},{id:"B68",body:'Fahmy MA. DRBEM sensitivity analysis and shape optimization of rotating magneto-thermo-viscoelastic FGA Structures using golden-section search algorithm based on uniform bicubic B-splines. Journal of Advances in Mathematics and Computer Science. 2017;25:1-20'},{id:"B69",body:'Fahmy MA. A predictor-corrector time-stepping DRBEM for shape design sensitivity and optimization of multilayer FGA structures. Transylvanian Review. 2017;XXV:5369-5382'},{id:"B70",body:'Fahmy MA. Shape design sensitivity and optimization for two-temperature generalized magneto-thermoelastic problems using time-domain DRBEM. Journal of Thermal Stresses. 2018;41:119-138'},{id:"B71",body:'Fahmy MA. Boundary element algorithm for modeling and simulation of dual-phase lag bioheat transfer and biomechanics of anisotropic soft tissues. International Journal of Applied Mechanics. 2018;10:1850108'},{id:"B72",body:'Fahmy MA. Modeling and optimization of anisotropic viscoelastic porous structures using CQBEM and moving asymptotes algorithm. Arabian Journal for Science and Engineering. 2019;44:1671-1684'},{id:"B73",body:'Fahmy MA. Boundary element modeling and simulation of biothermomechanical behavior in anisotropic laser-induced tissue hyperthermia. Engineering Analysis with Boundary Elements. 2019;101:156-164'},{id:"B74",body:'Fahmy MA, Al-Harbi SM, Al-Harbi BH, Sibih AM. A computerized boundary element algorithm for modeling and optimization of complex magneto-thermoelastic problems in MFGA structures. Journal of Engineering Research and Reports. 2019;3:1-13'},{id:"B75",body:'Fahmy MA. A new LRBFCM-GBEM modeling algorithm for general solution of time fractional order dual phase lag bioheat transfer problems in functionally graded tissues. Numerical Heat Transfer, Part A: Applications. 2019;75:616-626'},{id:"B76",body:'Fahmy MA. Design optimization for a simulation of rotating anisotropic viscoelastic porous structures using time-domain OQBEM. Mathematics and Computers in Simulation. 2019;66:193-205'},{id:"B77",body:'Fahmy MA. A new convolution variational boundary element technique for design sensitivity analysis and topology optimization of anisotropic thermo-poroelastic structures. Arab Journal of Basic and Applied Sciences. 2020;27:1-12'},{id:"B78",body:'Fahmy MA. Thermoelastic stresses in a rotating non-homogeneous anisotropic body. Numerical Heat Transfer, Part A: Applications. 2008;53:1001-1011'},{id:"B79",body:'Abd-Alla AM, Fahmy MA, El-Shahat TM. Magneto-thermo-elastic problem of a rotating non-homogeneous anisotropic solid cylinder. Archives of Applied Mechanics. 2008;78:135-148'},{id:"B80",body:'Fahmy MA, El-Shahat TM. The effect of initial stress and inhomogeneity on the thermoelastic stresses in a rotating anisotropic solid. Archives of Applied Mechanics. 2008;78:431-442'},{id:"B81",body:'Soliman AH, Fahmy MA. Range of applying the boundary condition at fluid/porous interface and evaluation of Beavers and Joseph’s slip coefficient using finite element method. Computation. 2020;8:14'},{id:"B82",body:'Eskandari AH, Baghani M, Sohrabpour S. A time-dependent finite element formulation for thick shape memory polymer beams considering shear effects. International Journal of Applied Mechanics. 2019;10:1850043'},{id:"B83",body:'Othman MIA, Khan A, Jahangir R, Jahangir A. Analysis on plane waves through magneto-thermoelastic microstretch rotating medium with temperature dependent elastic properties. Applied Mathematical Modelling. 2019;65:535-548'},{id:"B84",body:'El-Naggar AM, Abd-Alla AM, Fahmy MA, Ahmed SM. Thermal stresses in a rotating non-homogeneous orthotropic hollow cylinder. Heat and Mass Transfer. 2002;39:41-46'},{id:"B85",body:'El-Naggar AM, Abd-Alla AM, Fahmy MA. The propagation of thermal stresses in an infinite elastic slab. Applied Mathematics and Computation. 2003;12:220-226'},{id:"B86",body:'Abd-Alla AM, El-Naggar AM, Fahmy MA. Magneto-thermoelastic problem in non-homogeneous isotropic cylinder. Heat and Mass Transfer. 2003;39:625-629'},{id:"B87",body:'Hu Q, Zhao L. Domain decomposition preconditioners for the system generated by discontinuous Galerkin discretization of 2D-3T heat conduction equations. Communications in Computational Physics. 2017;22:1069-1100'},{id:"B88",body:'Huang FY, Liang KZ. Boundary element method for micropolar thermoelasticity. Engineering Analysis with Boundary Elements. 1996;17:19-26'},{id:"B89",body:'Eringen AC. Theory of micropolar elasticity. In: Liebowitz H, editor. Fracture. New York: Academic Press; 1968'},{id:"B90",body:'Dragos L. Fundamental solutions in micropolar elasticity. International Journal of Engineering Science. 1984;22:265-275'},{id:"B91",body:'Hoemmen M. Communication-Avoiding Krylov Subspace Methods. Berkeley: University of California; 2010'},{id:"B92",body:'Saad Y, Schultz MH. GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM Journal on Scientific and Statistical Computing. 1986;7:856-869'},{id:"B93",body:'Arnoldi WE. The principle of minimized iterations in the solution of the matrix eigenvalue problem. Quarterly of Applied Mathematics. 1951;9:17-29'},{id:"B94",body:'Lanczos C. An iteration method for the solution of the eigenvalue problem of linear differential and integral operators. Journal of Research of the National Bureau of Standards. 1950;45:255-282'},{id:"B95",body:'Pazera E, Jędrysiak J. Effect of microstructure in thermoelasticity problems of functionally graded laminates. Composite Structures. 2018;202:296-303'},{id:"B96",body:'Xiong QL, Tian XG. Generalized magneto-thermo-microstretch response during thermal shock. Latin American Journal of Solids and Structures. 2015;12:2562-2580'}],footnotes:[],contributors:[{corresp:"yes",contributorFullName:"Mohamed Abdelsabour Fahmy",address:"mohamed_fahmy@ci.suez.edu.eg",affiliation:'
Faculty of Computers and Informatics, Suez Canal University, Ismailia, Egypt
'}],corrections:null},book:{id:"10049",title:"Advanced Functional Materials",subtitle:null,fullTitle:"Advanced Functional Materials",slug:"advanced-functional-materials",publishedDate:"November 26th 2020",bookSignature:"Nevin Tasaltin, Paul Sunday Nnamchi and Safaa Saud",coverURL:"https://cdn.intechopen.com/books/images_new/10049.jpg",licenceType:"CC BY 3.0",editedByType:"Edited by",editors:[{id:"94825",title:"Associate Prof.",name:"Nevin",middleName:null,surname:"Tasaltin",slug:"nevin-tasaltin",fullName:"Nevin Tasaltin"}],productType:{id:"1",title:"Edited Volume",chapterContentType:"chapter",authoredCaption:"Edited by"}}},profile:{item:{id:"301586",title:"Prof.",name:"Colin",middleName:null,surname:"Snape",email:"colin.snape@nottingham.ac.uk",fullName:"Colin Snape",slug:"colin-snape",position:null,biography:null,institutionString:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",totalCites:0,totalChapterViews:"0",outsideEditionCount:0,totalAuthoredChapters:"1",totalEditedBooks:"0",personalWebsiteURL:null,twitterURL:null,linkedinURL:null,institution:null},booksEdited:[],chaptersAuthored:[{title:"Hybrid Two-step Preparation of Nanosized MgAl Layered Double Hydroxides for CO2 Adsorption",slug:"hybrid-two-step-preparation-of-nanosized-mgal-layered-double-hydroxides-for-co-sub-2-sub-adsorption",abstract:"Hybrid Two-step synthesis method for preparation of MgAl LDHs materials for CO2 adsorption has been employed because of the features of fast micromixing and enhanced mass transfer by using a ‘T-mixer’ reactor. MgAl LDHs with different morphologies were successfully obtained by three different synthesis routes: ultrasonication-intensified in ‘T-mixer’ (TU-LDHs), conventional co-precipitation (CC-LDHs) and ultrasonic-intensified in ‘T-mixer’ pretreatment followed by conventional co-precipitation (TUC-LDHs). The synthesized samples characterized by the XRD showed that LDHs formed a typical layered double hydroxide structure and no other impurities were identified in the compound. The SEM and TEM analyses also confirmed that the size distribution of TUC-LDHs was relatively uniform (with an average size of approximate 100 nm) and layered structure was clearly visible. The BET characterization indicated that such LDHs had a large surface area (235 m2 g−1), which makes it a promising adsorbent material for CO2 capture in practical application. It can be found that the CO2 adsorption capacities of TU-LDHs, CC-LDHs and TUC-LDHs at 80°C were 0.30, 0.22 and 0.28 mmol g−1, respectively. The CO2 adsorption capacities of TU-LDHs, CC-LDHs and TUC-LDHs at 200°C were 0.33, 0.25 and 0.36 mmol g−1, respectively. The order of CO2 adsorption capacity to reach equilibrium at 80°C seen in Avrami model is: TU-LDHs > TUC-LDHs > CC-LDHs. The CO2 adsorption/desorption cycling test reveals that TU-LDHs and TUC-LDHs have good adsorption stability than CC-LDHs.",signatures:"Xiani Huang, Xiaogang Yang, Guang Li, Collins I. Ezeh, Chenggong Sun and Collins Snape",authors:[{id:"234854",title:"Prof.",name:"Xiaogang",surname:"Yang",fullName:"Xiaogang Yang",slug:"xiaogang-yang",email:"Xiaogang.Yang@nottingham.edu.cn"},{id:"246362",title:"Dr.",name:"Guang",surname:"Li",fullName:"Guang Li",slug:"guang-li",email:"guang.li@nottingham.edu.cn"},{id:"292474",title:"Mrs.",name:"Xiani",surname:"Huang",fullName:"Xiani Huang",slug:"xiani-huang",email:"xiani.huang@nottingham.edu.cn"},{id:"301583",title:"Dr.",name:"Collins",surname:"Ezeh",fullName:"Collins Ezeh",slug:"collins-ezeh",email:"collins.ezeh@nottingham.edu.cn"},{id:"301585",title:"Dr.",name:"Chenggong",surname:"Sun",fullName:"Chenggong Sun",slug:"chenggong-sun",email:"cheng-gong.sun@nottingham.ac.uk"},{id:"301586",title:"Prof.",name:"Colin",surname:"Snape",fullName:"Colin Snape",slug:"colin-snape",email:"colin.snape@nottingham.ac.uk"}],book:{title:"Sorption in 2020s",slug:"sorption-in-2020s",productType:{id:"1",title:"Edited Volume"}}}],collaborators:[{id:"234854",title:"Prof.",name:"Xiaogang",surname:"Yang",slug:"xiaogang-yang",fullName:"Xiaogang Yang",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:null},{id:"246362",title:"Dr.",name:"Guang",surname:"Li",slug:"guang-li",fullName:"Guang Li",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:null},{id:"292474",title:"Mrs.",name:"Xiani",surname:"Huang",slug:"xiani-huang",fullName:"Xiani Huang",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:null},{id:"301583",title:"Dr.",name:"Collins",surname:"Ezeh",slug:"collins-ezeh",fullName:"Collins Ezeh",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:null},{id:"301585",title:"Dr.",name:"Chenggong",surname:"Sun",slug:"chenggong-sun",fullName:"Chenggong Sun",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:null},{id:"303694",title:"Prof.",name:"Ndubisi",surname:"Aviara",slug:"ndubisi-aviara",fullName:"Ndubisi Aviara",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:null},{id:"305169",title:"Distinguished Prof.",name:"Anna",surname:"Derylo-Marczewska",slug:"anna-derylo-marczewska",fullName:"Anna Derylo-Marczewska",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:null},{id:"307179",title:"Prof.",name:"André",surname:"Lajeunesse",slug:"andre-lajeunesse",fullName:"André Lajeunesse",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:null},{id:"307180",title:"Mr.",name:"Alexandre",surname:"Camiré",slug:"alexandre-camire",fullName:"Alexandre Camiré",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:{name:"Université du Québec à Trois-Rivières",institutionURL:null,country:{name:"Canada"}}},{id:"307181",title:"Prof.",name:"Bruno",surname:"Chabot",slug:"bruno-chabot",fullName:"Bruno Chabot",position:null,profilePictureURL:"//cdnintech.com/web/frontend/www/assets/author.svg",biography:null,institutionString:null,institution:{name:"Université du Québec à Trois-Rivières",institutionURL:null,country:{name:"Canada"}}}]},generic:{page:{slug:"partnerships",title:"Partnerships",intro:"
IntechOpen has always supported new and evolving ideas in scholarly publishing. We understand the community we serve, but to provide an even better service for our IntechOpen Authors and Academic Editors, we have partnered with leading companies and associations in the scientific field and beyond.
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OASPA
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The Open Access Scholarly Publishers Association (OASPA) was established in 2008 to represent the interests of Open Access (OA) publishers globally in all scientific, technical and scholarly disciplines. Its mission is carried out through exchange of information, the setting of standards, advancing models, advocacy, education, and the promotion of innovation.
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STM
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COPE
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Creative Commons
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Creative Commons (CC) is a nonprofit organization that enables the sharing and use of creativity and knowledge through free legal tools. IntechOpen uses the CC BY 3.0 license for chapters, meaning Authors retain copyright and their work can be reused and adapted as long as the source is properly cited and Authors are acknowledged.
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Crossref
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Altmetric and Dimensions from Digital Science
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DORA
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iThenticate
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Enago
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Amazon
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DHL
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The Association of Learned and Professional Society Publishers (ALPSP) is the largest association of scholarly and professional publishers in the world. Its mission is to connect, inform, develop and represent the international scholarly and professional publishing community. IntechOpen has been a member of ALPSP since 2016 and has consequently stayed informed about industry trends through connecting with peers and developing jointly.
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OASPA
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The Open Access Scholarly Publishers Association (OASPA) was established in 2008 to represent the interests of Open Access (OA) publishers globally in all scientific, technical and scholarly disciplines. Its mission is carried out through exchange of information, the setting of standards, advancing models, advocacy, education, and the promotion of innovation.
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STM
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The International Association of Scientific, Technical and Medical Publishers (STM) is the leading global trade association for academic and professional publishers. As a member, IntechOpen has not only made a commitment to STM's Ethical Principles.
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COPE
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The Committee on Publication Ethics (COPE) provides advice to editors and publishers on all aspects of publication ethics and, in particular, how to handle cases of misconduct in research and publication. IntechOpen has been a member of COPE since 2013 and adheres to the COPE Code of Conduct and Best Practice Guidelines, ensuring that we maintain the highest ethical standards.
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Creative Commons
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Creative Commons (CC) is a nonprofit organization that enables the sharing and use of creativity and knowledge through free legal tools. IntechOpen uses the CC BY 3.0 license for chapters, meaning Authors retain copyright and their work can be reused and adapted as long as the source is properly cited and Authors are acknowledged.
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Crossref
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Crossref is the official Digital Object Identifier (DOI) Registration Agency for scholarly and professional publications with a goal of making scholarly communications more effective. IntechOpen deposits metadata and registers DOIs for all content using the Crossref System. IntechOpen also deposits its references and uses the Crossref Cited-by service that enables researchers to track citation statistics.
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Altmetric and Dimensions from Digital Science
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Digital Science is a technology company serving the needs of scientific and research communities at key points along the full cycle of research. They support innovative businesses and technologies that make all parts of the research process more open, efficient and effective. IntechOpen integrates tools such as Altmetric to enable our researchers to track and measure the activity around their academic research and Dimensions, to ease access to the most relevant information and better understand and analyze the global research landscape.
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CLOCKSS
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CLOCKSS preserves scholarly publications in original formats, ensuring that they always remain available and openly accessible to everyone.
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Counter
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COUNTER provides the Code of Practice that enables publishers and vendors to report usage of their electronic resources in a consistent way. This enables libraries to compare data received from different publishers and vendors.
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DORA
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DORA is a worldwide initiative covering all scholarly disciplines which recognizes the need to improve the ways in which the outputs of scholarly research are evaluated and seeks to develop and promote best practice. To date it has been signed by over 1500 organizations and around 14,700 individuals.
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iThenticate
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Enago
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IntechOpen collaborates with Enago, through its sister brand, Ulatus, one of the world’s leading providers of book translation services. Their services are designed to convey the essence of your work to readers from across the globe in the language they understand.
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SPi Global
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Amazon
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Amazon is the world’s largest online retailer and cloud services provider. IntechOpen books have been available on Amazon since 2017, guaranteeing more visibility for our Authors and Academic Editors.
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DHL
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IntechOpen has partnered with DHL since 2011 to ensure the fastest delivery of Print on Demand books.
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